Kernel Bounds for Path and Cycle Problems

Kernel Bounds for P ath and Cycle Problems ∗ Hans L. Bo dlaender † Bart M. P . Jansen † Stefan Kratsc h † June 19, 2018 Abstract Connectivity problems like k -P a th and k -Disjoint P a ths relate to many impor tan t mile- stones in parameter ized complexity , namely the Graph Mino rs Pro ject, color co ding, and the recent developmen t of tec hniques for obtaining k er nelization low er b o unds. This w or k explores the existence of p olynomial k ernels for v arious path and cycle problems, b y co ns idering nonstan- dard pa rameterizations. W e show p olynomial kernels when the par ameters are a g iv en vertex cov er, a mo dulator to a cluster gr aph, or a (promised) max leaf num b er. W e o btain lo wer bounds via cros s-compo s ition, e.g., for H amil tonian Cycle and related pr oblems when para meterized by a mo dulator to an outer planar graph. k eywords: parameterized complexit y , k ernelization, upp er and lo wer b ounds , grap h s, path and cycle problems 1 In tro duction Connectivit y problems suc h as k -P a th and k -Di s joint P a ths pla y imp ortan t theoretical and practical r ol es in the ﬁ eld of parameterized complexit y . On the practical side, k -P a th [22, ND29] has applicat ions in computational biology [29] where the inv olv ed parameter is fairly sm al l, thus giving an excellen t opp ortunit y to apply parameterize d algorithms to ﬁ nd optimal solutio ns . On the theoretic al side, these problems ha v e triggered the dev elopmen t of v ery p ow erful alg orithmic tec hniques. The k -Disjoint P a ths prob lem [28] lies at the heart of th e Graph Minors Algorithm, and is the sour ce of t h e irr elevant-vertex tec hn ique. Th e c olor c o ding tec hnique of Alon et a l. [1] to solv e k -P a th has fou n d a wide range of applicatio ns and extensions [27, 9], and new m et h ods of solving k -P a th are still d ev eloping [3]. Despite the success stories of p arame terized algorithms for these problems, the quest for polynomial k ernels h as resulted in mostly neg ative results. Ind ee d , the failure to ﬁnd a p olynomial kernel for k -P a th w as one of the main motiv ations for the devel- opmen t of the kerneliza tion low er-b ound framew ork of Bo dlaender et al. [5 ]. Using th e framew ork it wa s sho wn that k -P a t h do es not admit a p olynomial kernel unless NP ⊆ coN P / p oly, ev en w h en restricted to v ery sp eciﬁc graph classes suc h as planar cubic graphs. It did n ot tak e long b efore re- lated connectivit y problems such as k -Disjoint P a t hs [7], k -Disjoint Cycles [7], k -Connec ted Ver tex Cover [17], and restricte d v arian ts of k -Connected Domina ting S et [14] we r e also sho wn not to admit p olynomial k ern el s unless NP ⊆ coNP / p oly. ∗ This w ork wa s supp orted b y the N etherlands O rga n iza tion for Scientiﬁc R esear ch (NWO), pro ject “KERNELS: Com binatorial An alysis of Data Reduction”. † Utrech t Univers ity , P .O. Bo x 80.089 , 3508 TB U trec ht, The Netherlands, { h.l.bodl aender,b.m.p.jans en ,s.k r atsch } @uu.nl 1 Th u s it seems t h at connectivit y requiremen ts in a pr oblem form a barrier to p olynomial ker- nelizabilit y w hen it comes to th e natural parameterizatio n by solution size k . Driv en by the desire to obtain useful prepro cessing pr ocedures for suc h pr ob lems, we ma y therefore in vestig ate the k er- nelization complexit y for non-standard parameters. Early w ork b y F ello ws et al. [20] sho ws that suc h a diﬀerent p ersp ecti ve can yield p olynomial k ernels: they prov ed that Hamil to nian Cycle parameterized by the max leaf n um b er of th e in put graph G , i.e. the maximum num b er of lea ves in a s p anning tree for G , admits a linear-v ertex ke r nel. In this wo rk we stud y the existence of p olynomial k ernels for v arious structur al paramet ers such as the max leaf n u m b er, the size of a v ertex co v er, an d the v ertex-deletion distance to simple graph classes suc h as cluster graphs and outerplanar graphs. O ur results are as follo ws 1 : 1. W e int r odu ce a widely applicable tec hnique based on matc hings in bipartite graphs to sh o w that the problems Lo ng Cycle , its directed and path v ariants, Disjoint P a ths , and Dis- joint Cyc les , admit k ernels with O ( | X | 2 ) vertice s when parameterized b y a vertex co ver X . 2. W e generalize these results to the str on ger (i.e. smaller) parameter “ve rtex-deletion distance to a cluster graph”. F or Long Cycle a n d Long P a th this requ ires the t r ic k o f enco ding clique sizes in bin ary (in a weigh ted v ariant) and app lying a Karp reduction to get “bac k” to the origi n al problem. On the one hand, th is h as the drawbac k of not giving an exp lic it p olynomial size b oun d. On th e other hand, t h e binary enco ding of cl iqu e sizes seems to b e more eﬃcien t for s u bsequen t computations. F or Disjoint Cycles and Disjoint P a ths the Karp redu ction can b e a v oided, as the length of cycles or paths is not imp ortant . 3. Using the same binary enco ding tr ic k we get p olynomial kernels for Lon g Cycle and Long P a th parameterized by th e max leaf num b er, generalizing the r esu lt of F ello ws et al. [20] for Hamil t onian Cycl e . F or Disjoint Cycles and Disjoint P a th s the enco ding trick is again not necessary . 4. W e giv e contrasting k ernelization low er b ounds using the recen tly introdu ced tec hnique of cross-comp ositio n [6]: ( a) Directed Hamil tonian Cycle p arameterized by a m odu- la t or to Bi-p a ths do es not admit a p olynomial k ern el unless NP ⊆ coNP / p oly, where the parameter measures the v ertex-deletio n distance to a dig r aph whose und erlying undirected graph is a path, and (b) w e mod ify the construction to pro v e that Hamil to nian Cycle p a- rameterize d by a modula tor to Outerpl anar graphs do es not admit a p olynomial k ernel; b oth r esu lts assum ing NP 6⊆ coNP / p oly. 5. W e in iti ate the parameterized complexit y stud y o f ﬁ n ding paths resp ect in g forbidden pairs [22, GT54] under v arious parameteriza tions. W e obtain W[1]-hardness pro ofs, FPT algorithms, k ernel lo w er b ounds, and para-NP-completeness resu lts. Related wo rk. Chen and Flum show ed that Maximal k -P a th is in FPT, and that Maximal Induced k -P a th is W[2]- complete [10]. Joined b y M ¨ uller they studied v arious f orm s of ke rn eliz a- tion lo wer b ounds, and s ho w ed amongst others that k -Poi nt ed P a th do es not admit a parameter non-increasing p olynomial k ernelization u nless P = NP , and that k -P a th do es n ot hav e a p olyno- mial k ern el on connected planar graphs unless NP ⊆ coNP / poly [11]. 1 It is easy to see that Hami l tonian Cycle and Ham il tonian P a th are sp ecial cases of Long Cycle and Long P a th respectively . Hence, upp er and lo wer boun ds transfer in the ob vious w ay b etw een them. 2 Organization. Sectio n 2 introdu ces the necessary notation regarding graphs and parameterized complexit y , a nd introd uces the main problems studied in th is w ork. In Section 3 w e sh o w a useful prop ert y of bipartite matc hin gs, which helps to obtain k ernelization resu lts. Section 4 con tains the p ositiv e results, i.e., p olynomial k ern els, for the men tioned p at h and cycle problems w hen parameterized b y a vertex co ver (Section 4.1), the max leaf num b er (Section 4.2), or a mo dulator to a cluster graph (Section 4.3). In Section 5 w e sh ow th e mentio ned lo we r b ound results f or directed a n d undir ec ted Hamil to nian Cycle . Section 6 con tains our results for path p roblems with forbidd en pairs. W e conclude in S ec tion 7. 2 Preliminaries Graphs. Al l graphs are ﬁnite and simple, unless in dicat ed otherwise. An un directed graph G has a ve rtex set V ( G ) an d an edge set E ( G ) ⊆  V ( G ) 2  . A directed graph D h as a v ertex set V ( D ) and a set of directed arcs A ( D ) ⊆ V ( D ) 2 . All paths are assumed to b e simple. W e sa y that a matc hing M in a graph c overs a s et of v ertices U , if eac h vertex in U is endp oin t of an edge in M . W e u s e [ n ] as a shorthand for the set { 1 , 2 , . . . , n } . If X is a ﬁ nite set then  X n  denotes the set of all size- n subs et s of X . F or a directed graph D and vertex v w e write N + D ( v ) := { u | ( v , u ) ∈ A ( D ) } for the out- neigh b ors, N − D ( v ) := { u | ( u, v ) ∈ A ( D ) } for the set of in-neigh b ors, and N D ( v ) := N + D ( v ) ∪ N − D ( v ) for the set of all neighbors. If ( u, v ) is an arc of D th en u is th e he ad of the arc and v is its tail . If C ⊆ A ( D ) is a Hamiltonian cycle in a digraph con taining the arcs ( x i , x i +1 ) f or i ∈ [ k ] then we sa y that th e v ertices x 1 , . . . , x k +1 app ear consecutiv ely on C , in that ord er. W e also use the undirected v ariant of this notion wh ic h is d eﬁned analogo u sly . F or a set of ve rtices X in a digraph D w e use D − X to d enote the digraph wh ic h results after deleting all vertices of X and their inciden t arcs from D ; the concept is deﬁned analogously for u n directed grap h s. The underlying undirected graph of a d ig r aph D is the result o f disregarding the orien tation of the arcs and eliminating parallel edges. Let Bi-p a ths (for bi-orient ations of paths) b e the class of d igraphs whose und erlying undirected grap h is a p at h . Ou terp lanar gr aphs are those graphs whic h can b e dr a wn in the plane w ithout crossings su c h that all the vertices lie on the ou ter face; suc h graphs h a v e treewidth at most t wo. Cluster gr aphs are d isj oi nt unions of cliques. F or a graph class G and a verte x set X ⊆ V ( G ) of a graph G such that G − X ∈ G , w e sa y that X is a mo dulator to the class G . P arameterized c omplexity and kernelization. A parameterized p roblem Q is a s u bset of Σ ∗ × N , the second comp onent b eing the p ar ameter whic h exp r esses some structural m ea su re of the input. A parameterized pr oblem is (strongly uniformly) ﬁxe d-p ar ameter tr actable if there exists an algorithm to d ec ide wh ether ( x, k ) ∈ Q in time f ( k ) | x | O (1) where f is a computable function [18]. A kernelization algorithm (or kernel ) for a parameterized problem Q is a p olynomial-ti me algorithm whic h transf orms an instance ( x, k ) in to an equiv alent instance ( x ′ , k ′ ) su c h that | x ′ | , k ′ ≤ f ( k ) for some computable function f , whic h is the size of the ke rn el. If f is a p olynomial then this is a p olyno mial kernel [23]. Cross-comp osition. T o pro ve our lo w er b ounds w e use the framewo r k of cross-comp ositio n [6], whic h b uilds on earlier wo rk b y Bod la end er et al. [5], and F ortn ow and S an thanam [21]. 3 Deﬁnition 1 (Po lynomial equiv alence rel ation [6]) . An equiv alence relatio n R on Σ ∗ is calle d a p olynomial e qu ivalenc e r elation if the follo wing t wo conditions h old: 1. Th ere is an algo rith m that giv en t w o strings x, y ∈ Σ ∗ decides wh ether x and y b elong to the same equiv alence class in ( | x | + | y | ) O (1) time. 2. F or any ﬁ nite set S ⊆ Σ ∗ the equiv alence relation R partitions th e elemen ts of S into at most (max x ∈ S | x | ) O (1) classes. Deﬁnition 2 (Cr oss-co mp osition [6]) . Let L ⊆ Σ ∗ b e a set and let Q ⊆ Σ ∗ × N b e a p aramet erized problem. W e sa y that L cr oss-c omp oses into Q if there is a p olynomial equiv alence relatio n R an d an algorithm wh ic h, giv en r strings x 1 , x 2 , . . . , x r b elonging to the same equiv alence class of R , computes an instance ( x ∗ , k ∗ ) ∈ Σ ∗ × N in time p olynomial in P r i =1 | x i | such th at : 1. ( x ∗ , k ∗ ) ∈ Q ⇔ x i ∈ L for some 1 ≤ i ≤ r , 2. k ∗ is b ounded b y a p olynomial in max r i =1 | x i | + log r . Theorem 1 ([6]) . If some set L ⊆ Σ ∗ is NP-har d under K ar p r e ductions and L c r oss-c omp oses into the p ar ameterize d pr oblem Q then ther e is no p olyn omial kernel for Q unless NP ⊆ c oNP / p oly. Problems considered in this work. The main fo cus of this w ork lies on nonstandard parame- terizatio n s of Lon g P a th , Long C yc le , Hamil tonian P a th , Hamil t onian Cyc le , Disjoint P a ths , and Disjoint Cycles . W e brieﬂy in tro duce t h e classical un p aramete rized ve rs io n s; the ﬁrst four a quite w ell known: Giv en a graph G and an integ er k , Long P a th and Long Cycle ask for the existe n ce of a p ath or cycle r espective ly conta inin g at least k ve r tic es. Giv en a g r aph G , Hamil t onian P a th and Hamil tonian Cycle ask for the existence of a path or cycle resp ectiv ely whic h conta in s all ve rtices of G . Disjoint P a ths Input: A graph G , an inte ger k , and a set of k v ertex pairs { ( s 1 , t 1 ) , . . . , ( s k , t k ) } . Question: Is there a set of k v ertex-disjoint paths, su c h that eac h p air ( s i , t i ) is con- nected by exactly one of the paths? Disjoint Cycl es Input: A graph G and an inte ger k . Question: Do es G con tain a set of k v ertex-disjoint cycles? Most of the v arian ts with n onstandard parameters considered in this work are giv en b y requirin g an extra set X (a mo dulator) to b e giv en in the inp ut such that G − X has some sp ecial prop ert y (lik e b eing an indep endent set). The p aramet er is then c hosen as ℓ := | X | . F or formal reasons this w ould r equire an explicit inclusion of ℓ in the inpu t, whic h we omit for succinctness. F u r ther v ariants of path problems w ith forb idden pairs are in tro duced in Section 6. 3 A prop ert y of maxim um matc hings in b iparti te graphs The f ol lowing theorem simpliﬁes th e argumen tation needed for reduction ru les that are based on assigning priv ate c hoices to some en tities, e.g., v ertices or edges, u sing an auxiliary bipartite graph, useful for example f or v arious problems p aramete rized b y a v ertex co v er. 4 Theorem 2. L et G = ( X ∪ Y , E ) b e a bip artite gr aph. L et M ⊆ E ( G ) b e a maximum matching in G . L et X M ⊆ X b e the set of vertic es in X that ar e endp oint of an e dge in M . Then, for e ach Y ′ ⊆ Y , if ther e exists a matching M ′ in G th at c overs Y ′ , then ther e exists a matching M ′′ in G [ X M ∪ Y ] that c overs Y ′ . Pr o of. Let G , M , and X M b e as stated in the theorem. Supp ose the theorem d o es not hold for Y ′ ⊆ Y , and let M ′ b e a matc hin g in G that co v ers Y ′ . Over all suc h matc hings M ′ , take one that co ve r s the largest num b er of ve r tic es in X M . By assump tio n M ′ is not a m atching in G [ X M ∪ Y ], so there is a ve r tex y 0 ∈ Y ′ that is matc hed in M ′ to a v ertex in X \ X M , s ay x 0 . W e use an iterati ve pro cess to deriv e a con tradiction, m aintaining the follo wing in v arian ts: • x 0 6∈ X M . • { x j , y j } ∈ M ′ for 0 ≤ j ≤ i . • { y j , x j +1 } ∈ M f or 0 ≤ j < i . • The vertice s x j for 0 ≤ j ≤ i are distinct members of X , and ve rtices y j for 0 ≤ j ≤ i are distinct members of Y . It is easy to ve r if y that giv en our c hoice of x 0 , y 0 these in v arian ts are initially satisﬁed for i = 0. W e no w contin ue th e pr ocess based on a case distinction: 1. If y i is not matc hed under M , then the s equ ence ( x 0 , y 0 , . . . , x i , y i ) is an M -augmenting p ath in G since x 0 and y i are not m at ched un der M , and all edges { y j , x j +1 } for 0 ≤ j < i are con tained in M . He n ce M ′′ := M \ { { y j , x j +1 } | 0 ≤ j < i } ∪ {{ x j , y j } | 0 ≤ j ≤ i } is a matc hing in G larger than M , con trad icting that M is maximum. 2. In the r emai n ing cases we ma y assume y i is matc hed un der M , s ay { y i , x i +1 } ∈ M . If there is an index 0 ≤ j ≤ i su c h that x i +1 = x j then j > 0 (since x 0 6∈ X M ) and the edges { y i , x i +1 } and { y j − 1 , x j } are b oth con tained in M and are distinct edges since y j − 1 6 = y i , con tradicting the fact that M is a matching. Hence x i +1 is distinct from x j for 0 ≤ j ≤ i . 3. If x i +1 is not co vered b y M ′ then th e matc hing M ′′ := M ′ \ {{ x j , y j } | 0 ≤ j ≤ i } ∪ {{ y j , x j +1 } | 0 ≤ j ≤ i } con tains as many edges as M ′ but co v ers more v ertices of X M , con tradicting the c hoice of M ′ . Hence x i +1 is co vered by M ′ , say { x i +1 , y i +1 } ∈ M ′ . If there is an index 0 ≤ j ≤ i suc h that y i +1 = y j then { x i +1 , y i +1 } and { x j , y j } are tw o distinct edges in M ′ inciden t on y i +1 , con tradicting that M ′ is a matc hing. Hence y i +1 is d istinct from y j for 0 ≤ j ≤ i . No w observe that the inv ariant h olds for i + 1, and w e may pro ceed with the next step of th e pr ocess. By the last prop ert y of the inv arian t, the pro cess m ust end . Hence the assumption that there is no matc hing in G [ X M ∪ Y ] w hic h co v ers Y ′ leads to a con tradiction, which concludes the pro of. 4 P olynomial k ernels for path and cycle problems This section pro vides p olynomial ke r nels for v arious path and cycle problems wh en parameterize d b y a ve r te x cov er (Section 4.1), the max leaf num b er (Section 4.2), or a mo dulator to a cluster graph (Section 4.3). 5 4.1 P arameterization b y a v ertex cov er In this section we consider path and cycle p roblems when parameterized by the size ℓ of a given v ertex co v er, f ocusing mainly on th e Long C y cle pr ob lem, for whic h w e p resen t a k ern el w ith O ( ℓ 2 ) v ertices. Long Cycle p arameter ized by a ver tex cover Input: A graph G , an inte ger k , and a set X ⊆ V ( G ) s uc h that X is a v ertex co v er. P arameter: ℓ := | X | . Question: Do es G con tain a cycle of length at least k ? W e need only o n e redu ctio n rule to get a k ernelizatio n, it uses a bip artite c onne ction gr aph H = H ( G, k , X ): On e color class consists of the v ertices in the indep endent set I = V ( G ) \ X , and the other consists of a ll (unordered) pairs of d istinct v ertices in X . W e tak e an edge fr om a vertex v ∈ I to a ve rtex represent ing the pair { p, q } ⊆ X , if and only if v is adjacen t to p and to q . Reduction Rule 1. Giv en ( G, k , X ), if k ≤ 4 then solv e the p roblem (e.g. by th e trivial O ( n 4 ) algo- rithm) and return an equiv alen t dummy instance. Otherwise, construct th e connection graph H = H ( G, k , X ). L et M b e a maximum matc h in g in H . Let J ⊆ I b e the ve r tices co v ered by an edge in M . Remov e all v ertices in I \ J and th eir incident edges from G . Let G ′ b e the resulting graph, and return the in stance ( G ′ , k , X ). Observ at ion 1. In Rule 1, | J | is at most the n umber of p airs of distinct ve rtices in X , and hen ce after applying the rule, G ′ has at most ℓ +  ℓ 2  ∈ O ( ℓ 2 ) v ertices. Correctness of the rule follo ws fr om the follo wing lemma. Lemma 1. L et ( G, k , X ) b e an instanc e of Long Cycle p arameterize d by a ver tex cover , and let ( G ′ , k , X ) b e the instanc e r eturne d by R ule 1. Then G has a cycle of length at le ast k if and only if G ′ has a cycle of length at le ast k . Pr o of. If k ≤ 4 then the lemma holds trivially . Otherw ise, w e ha ve that G ′ is an induced s u bgraph of G so cycles (in particular th ose of length at least k ) in G ′ exist also in G . I t r emai n s to lo ok at the con verse. Let C b e a cycle of length at least k ≥ 5 in G . Clearly , as I = V \ X is an indep endent set, any v ertices of I whic h are in C must b e neigh b ored b y v ertices of X on C . Let v 1 , . . . , v r b e all ve r tices of I con tained in C and let p i and q i b e the predecessor and successor of v i on C , resp ectiv ely (clearly r ≤ ℓ b ut there migh t b e far fewe r v ertices of I on C ). Sin ce C has length at least 5, it follo ws that { p i , q i } 6 = { p j , q j } f or all i, j ∈ [ r ] with i 6 = j (else it w ould hav e length 4). T o s h o w that G ′ con tains a cycle of length at least k , it suﬃces to ﬁnd replacemen ts for all ve rtices v i whic h are not in J (and h ence not in G ′ ); for this we will use the matc h ing. Clearly , in H = H ( G, k , X ) there is a matc hing M co v ering W := {{ p 1 , q 1 } , . . . , { p r , q r }} , namely matc hing eac h pair to th e corresp onding verte x v i . F urther, by Rule 1, J is the set of endp oin ts in I of some maxim um matc hin g of H . Hence, by Theorem 2, there is a matc hing M ′ co v ering W in H [ J ∪ W ]. Let v ′ i denote the vertex matc hed to { p i , q i } by M ′ , for i ∈ { 1 , . . . , r } . It is easy to see that w e ma y replace eac h v i on C by v ′ i since v ′ i is adjacen t to p i and q i in G , obtaining a cycle C ′ whic h in tersects I only in ve r tic es of J . Also, as all pairs { p i , q i } are diﬀerent, no vertex v ′ i is requir ed t wice. Hence, C ′ is also a cycle of G ′ , and of length at least k . 6 The k ernelization result no w follo ws from Lemma 1 and Observ ation 1, and noting that Rule 1 can b e easily p erformed in p olynomial time. Theorem 3. Long Cycle p arameterized by a ver tex cover has a kernel with O ( ℓ 2 ) ve rtic es. W e note that in the obtained kernel the n umb er of edges may still b e cubic in ℓ , giving an o v erall s ize b ound of O ( ℓ 3 ) by using an adjacency matrix enco ding. It w ould b e inte r esti n g to know whether the n umber of ed ge s can b e redu ced to O ( ℓ 2 ) and wh ether a size b ound of O ( ℓ 2 ) could b e sho wed to b e tight, using recen t results on p olynomial lo wer b ounds for k ern eli zation [16, 15, 25]. F urther problems para meterized b y v ertex co ver. Th e same tec hn ique can b e used for a n umb er of additional p roblems, all p arame terized by the size of a v ertex co ver. T he b asic argumen t is the same; the matc hing approac h allo ws us to r eroute an y paths or cycles su c h that they u se only matc hed v ertices. Corollary 1. Long P a t h , Disjoint P a ths , and Disjo int Cycles admit p olynomial kernels with O ( ℓ 2 ) vertic es. Pr o of. W e brieﬂy sketc h how to mod ify the pro of given for Long Cycle (Theorem 3). F or an instance ( G, k , X ) of th e Long P a th problem, it is most con venien t to introd uce a unive rs al v ertex v (adjacen t to all v ertices of G ) to obtain an equ iv alen t instance ( G ′ , k + 1 , X ∪ { v } ) of Long Cycle parameterized b y a vertex co ver. E ac h k -path in G then corresp onds to a k + 1- cycle in G ′ b y adding v , and eac h k + 1-cycle in G ′ con tains at le ast one k -p at h in G . W e then apply the ke r n eliz ation as for Long Cycle , and ﬁnish up by removing v from the obtained instance. F or Disjoint P a ths it is clear that eac h path f u lﬁlling a requ est ( s i , t i ) must conta in at least one ve r tex of X , hence ye s -instances h a v e at most ℓ = | X | request p airs. Thus w e ma y assume that all v ertices of request p airs are con tained in X (else adding them will at most triple the size of X ). Since the paths hav e to b e v ertex-disjoin t, no tw o request pairs share an y v ertices (though the pro of could b e adapted to int ern all y v ertex-disjoin t p aths, and also to asking for m ultiple paths b et w een certain pairs). It is hence clear that edges b et ween ve r tices of diﬀerent p airs cann ot b e part of an y path. T herefore, th e easiest wa y to argue correctness is to add all those edges to G , and observe that the k ern eliz ation f or Long Cycl e also preserv es th e existence of a cycle whic h tra v erses all request pairs in order, i.e., ( . . . , s 1 , . . . , t 1 , s 2 , . . . , t 2 , s 3 , . . . ). Suc h a cycle exists if and only if the k requested disjoint paths exist and the instance is yes . F or the Disjoint Cycles prob lem we may pro ceed essent ially as for Long Cycle , since eac h single cycle will b e p reserv ed (as the argument only comes down to pr oviding the p r iv ate shared neigh b ors). T he only diﬀerence is that we also h av e to preserve cycles of length four, whic h ma y require t wo vertic es of the in dep enden t set to b e assigned (matc hed ) to one pair of v ertices of the vertex co ver X . (W e also must preserve cycles of length thr ee of cours e, bu t only length four cycles ma y ha ve the p artic u la r mentioned la yo u t.) T o d o so create the auxiliary bipartite graph H = H ( G, k , X ) bu t dup lica te all vertice s corresp onding to (unordered) pairs of vertic es from X . T his w a y , eac h pair can receiv e tw o priv ate shared neigh b ors. F or Hamil tonian P a th and Ha mil tonian Cycle it is easy to s ee that an y v ertex co v er of a yes -i n stance m us t ha ve size at least least ⌊ | V | 2 ⌋ , since vertice s of the remaining indep endent set cann ot b e ad j ac ent on a Hamiltonian path or cycle. Thus all non trivial instances ( G, X ) ha ve | V ( G ) | ≤ 2 | X | + 1 = O ( ℓ ). 7 F u r thermore, the matc hing argumen t can also b e applied to the directed v ersions of these problems. F or this, matc h to eac h ordered p air ( u, v ) with u and v b eing v ertices of the provided v ertex co ve r a verte x p such that there are (directed) edges ( u, p ) and ( p, v ) (note that there will b e a vertex q for ( v , u ) as w ell, with edges ( v , q ) and ( q , u )). This w a y , an y directed paths or cycles can b e rerouted to use only m at ched vertic es fr om V ( G ) \ X , pro vidin g kernels of the same size (up to constan t factors). Again, for Ham i l tonian P a th and Hamil tonian Cycle the simpler argumen t from the previous paragraph app lies. 4.2 P arameterization b y max leaf n umber In this section we consider path and cycle problems parameterized by the max leaf num b er, i.e., the maxim um n umb er of lea v es in an y spanning tree of the graph. Deviating sligh tly from the standard use, we will tak e the max leaf num b er of a disconnected graph to b e the sum of max leaf num b ers tak en o ver all connected comp onen ts (alternativ ely one ma y restrict the question to connected graphs). W e will use Long Cycle as a runn in g example, b ut a s in Section 4.1 it is easy to generalize th e arguments to fur th er prob lems. As the max leaf n u m b er of a graph cannot b e v eriﬁed in p olynomial time, w e consider the parameterization in the s en se of a promise problem, e.g.: Long Cycle p arameter ized by max leaf number (LCML) Input: A graph G and t wo in tegers k and ℓ . P arameter: ℓ . Question: If G has max leaf num b er at most ℓ , then decide whether G contai n s a cycle of length at least k . Else the output m ay b e arbitrary . It is well kno wn that a large graph ha ving small max leaf num b er must con tain long paths of degree tw o v ertices and few v ertices of degree at least th ree. The follo win g b ound was obtained b y F ello ws et al. [20] b ase d on w ork by Kleitman and W est [26]; it can b e easily seen to hold for eac h connected comp onen t. Lemma 2 ([20]) . If a gr aph G has max le af numb er at most ℓ then it is a sub division of some gr aph H of at most 4 ℓ − 2 vertic es. In p articular, G has at most 4 ℓ − 2 vertic es of de gr e e at le ast thr e e. It is not h ard to d evise an FPT-algo rith m for L C ML. Lemma 3. Long Cycle p arameterized by max leaf numbe r c an b e solve d in time 2 O ( ℓ ) n c . Pr o of. Let ( G, k , ℓ ) b e an instance of LC ML. Let B denote those vertices that ha ve degree at least three in G . If | B | > 4 ℓ − 2 then by Lemma 2, the promise is not fulﬁlled as the m ax leaf n umb er of G is greater than ℓ , w e retur n no . Other w ise, we ha v e | B | ≤ 4 ℓ − 2. No w, replacing eac h path connecting t wo B ve rtices (without visiting other B v ertices) by an edge with wei ght equal to the length of the path we obtain a m ultigraph with th e same m axim um cycle length. Clearly , a longest cycle will either consist of j u st t wo v ertices, or us e at most one of the edges b etw een an y t wo ve rtices. Hence, after chec king wh ether there is a su ﬃ cie ntly long 2- v ertex cycle, w e ma y d iscard all b ut the longest edge connecting an y t wo ve rtices. The remaining problem can b e solv ed b y Held-Karp style dynamic p r og r amming [24] on the remaining w eight ed graph with | B | = O ( ℓ ) v ertices; hence in time 2 O ( ℓ ) n c . 8 The main idea for the k ernelization is th at one of t wo goo d cases must hold: Either all the path lengths are small enough suc h that a binary enco ding of their length has size p olynomial in ℓ , or the total num b er n of v ertices is large enough such that the 2 O ( ℓ ) n c is in fact p olynomial in n . Theorem 4. Long Cycle p arameterized by max le af numb er admits a p olynomial kernel. Pr o of. Giv en an instance ( G, k , ℓ ) of LCML, we ﬁ rst c h ec k that k do es not exceed the num b er of v ertices and that there are at most 4 ℓ vertice s of degree at least three, or else return no . If G has more than 2 O ( ℓ ) v ertices (usin g the concrete b ound resulting f rom an implementat ion of Lemma 3), then we solve th e instance in time 2 O ( ℓ ) · n c = O ( n c +1 ), an d answer yes or no accordingly . Oth erwise let B denote the set of vertice s of degree at least th ree. If there are more than ℓ d isj oi nt paths connecting any tw o vertic es of B , then the max leaf num b er of G exceeds ℓ , and w e retur n no . W e rep lace eac h path connecting t wo v ertices b, b ′ ∈ B , with internal vertic es fr om V ( G ) \ B , b y a single edge with an integer lab el den oting the num b er of inte r nal v ertices of the replac ed path. W e obtain a m ultigraph G ′ in whic h some edges ha ve an in teger lab el. It is easy to see that cycles in G corresp ond to cycl es in G ′ of the same length, when taking the inte ger lab els in to acco u n t (i.e. lab eled edges are simply w orth as muc h as that many inte r n al vertic es). Clearly , eac h lab el can b e enco ded in binary by at m ost log 2 O ( ℓ ) = O ( ℓ ) bits. F urthermore, we delete all paths that start in a v ertex of B , hav e in ternal vertic es from V ( G ) \ B , and end in a v ertex of degree one; clearly those cannot b e used b y cycles in G . W e obtain a m ultigraph G ′ with at most 4 ℓ vertic es in B and with at most ℓ edges b etw een an y t wo B -ve r tices. Thus w e h a v e O ( ℓ ) ve r tic es and O ( ℓ 3 ) in teger lab els of s ize O ( ℓ ), for a total size of O ( ℓ 4 ) (this could b e easily tightened, b ut it w ould not aﬀect the result); clearly k can also b e enco ded in O ( ℓ ) bits. W e obtain an equiv alen t instance of a slightly diﬀerent Lon g Cycle problem on m ultigraphs in wh ic h s ome edges ma y b e lab eled, b u t whic h is in NP . By the imp lie d Karp reduction to LCML w e obtain the claimed p olynomial k ern el (cf. [7]). Deviating from Bo dlaender et al. [7] w e do not use the v ersions with parameter en co ded in unary , but observe the f ollo win g: All in sta n ce s of Long Cycle with k exceeding the num b er of v ertices are trivially no and m ay b e r eplac ed b y smaller dummy no -instances, so the parameter v alue of the remaining in s ta n ce s is ind ee d p olynomial in ℓ (as is the instance size, due to the K arp r ed uctio n ). F urther problems parameterized by max leaf num ber. A p olynomial k ern el for Hamil- tonian Cycle was already found b y F ello w s et al. [20]. Kernels for Hamil tonian P a th as well as Disj oint Cycle s can b e ob tained in a similar w a y , by observing that the p at h s of degree-2 v ertices can b e r educed to h a ving only one inte r nal v ertex. F or Long P a th it is again n ece ssary to use the b inary enco ding trick for the path lengths. Corollary 2. Long P a th , Ha mil tonian P a th , and Disjoint Cycles p ar ameterize d by max le af numb er admit p olynomial kernels. F or Disjoint P a t hs , i.e., ﬁ nding k disjoint paths connecting k terminal p airs ( s 1 , t 1 ) , . . . , ( s k , t k ), some more w ork is necessary on the p at h s b et wee n B v ertices, and on paths b et w een B vertic es and the at most ℓ leav es. Theorem 5. Disjoint P a ths p ar ameterize d b y max le af numb er admits a p olynomial kernel. 9 Pr o of. Let ( G, k , { ( s 1 , t 1 ) , . . . , ( s k , t k ) } , ℓ ) b e an instance of Disjoint P a ths p a rameterized by max l eaf numbe r (DPML). Let L b e the set of lea ve s of G and let B con tain all vertices of degree at least th ree. If | L | > ℓ or if | B | > 4 ℓ we r etur n no as, by Lemma 2, the max leaf n u m b er of G m ust exceed k . Similarly , we return no if an y v ertex of B has degree greater than ℓ . W e will now reduce the num b er of terminals on paths w ith internal v ertices from O := V ( G ) \ ( L ∪ B ), i.e., the set of v ertices with degree exactly t wo. Let P b e su ch a path and assume that it con tains at least ﬁ v e terminals. If follo ws that any terminal except the ﬁrs t and the last one on the p at h m u st reac h its p artner (e.g., s i m ust r ea ch t i ) on the path P . This can b e tested in p olynomial time, and th e in s ta n ce b e rejected if it is imp ossible. Afterw ards th e subpath spann ed b y those termin als can b e deleted and b e replaced by a subpath consisting jus t of the v ertices of one new terminal pair, sa y ( s ′ , t ′ ). By the same argumenta tion s ′ and t ′ m ust b e connected on the path, hence the path cannot b e u sed for other terminals. T o see that the max leaf n um b er do es not increase, observe that the ab o ve mo diﬁcation is basically a con traction of some edges (terminals ha ve no inﬂuence), and that it do es not increase the n u m b er of connected comp onen ts. Otherwise, w e r eplac e all non-terminal v ertices in O b y deleting them and add in g an edge b et w een their n eig hb ors; it is clear that an y path must p ass through th em, so this is equiv alen t. Afterw ards, eac h path with internal ve r tices from O has at most four internal v ertices, allo wing us to b oun d the total num b er of vertic es by | L | + | B | + 4 ℓ | B | ∈ O ( ℓ 2 ) (the latter term account s for the at most ℓ paths starting in any vertex of B ). 4.3 P arameterization b y a mo dula t or to cluster graphs In this section, w e co n sider p at h and cycle pr oblems parameterized by v ertex-deletion distance from cluster graphs. T o this end, alongside the input grap h (and p ossibly further inputs) a mo dulator X is pro vided suc h that G − X is a cluster graph. Again we consider the Long Cycle problem. Long Cycle p arameter ized by a modula tor to cluste r graphs Input: A graph G , an integ er k , and a set X ⊆ V ( G ) suc h that G − X is a disjoint union of cliques. P arameter: ℓ := | X | . Question: Do es G con tain a cycle of length at least k ? Observ at ion 2. Cycles in G conta in vertic es of at most | X | cliques of G − X . If a cycle uses at least one ed ge b et we en vertic es of a clique, then it can b e easily extended to include all so far unused v ertices of the clique. Hence, it can b e seen that there is a preference for including th e largest p ossible cliques and as many cliques as p ossible. Th is is used in the follo wing reduction rule. Reduction Rule 2. Giv en ( G, k , X ), if there is a clique with at least k ve rtices, or a v ertex in X with tw o neigh b ors in a clique of size k − 1 then delete all other cliques and return the obtained instance ( G ′ , k , X ) (whic h is trivially yes ). Otherwise, for ea ch vertex pair { u, v } from X , m ark ℓ + 1 cliques that con tain a shared neigh b or of u and v . Additionally , mark the ℓ + 1 largest cliques con taining v ertices p and q with u adjacen t to p and v adjacent to q . Delete all u n mark ed cliques, obtaining a graph G ′ , an d return ( G ′ , k , X ). Clearly G ′ − X is still a cluster graph, and if G ′ con tains a cycle of length at least k , then that cycle can also b e found in its sup ergraph G . Th e follo wing lemma completes the pro of for safeness of Rule 2. 10 Lemma 4. L et ( G ′ , k , X ) b e obtaine d fr om an applic ation of Rule 2 on ( G, k , X ) . If G has a c ycle of length at le ast k , then G ′ has a cycle of length at le ast k . Pr o of. Assume that G con tains a cycle of length at least k . If the ﬁrst part of Ru le 2 applies, then G ′ trivially conta ins a cycle of length k and w e are done. Otherwise, an y cycle of length at least k must contai n at least t wo v ertices of X . Let C b e a cycle of length at least k in G which con tains a minimum num b er of subpaths that are not contai n ed in G ′ − X . Ass ume that C conta ins ve r tices from at least one clique whic h is not in G ′ − X (i.e. which w as not marked) and let K b e su ch a clique of G − X . Let P = ( p 1 , . . . , p r ) denote one of the subp aths obtained b y intersecting C w ith the v ertex s et of K . F urther, let P uv denote th e extended sub path obtained by addin g the vertice s u, v ∈ X whic h are adjacen t to P on C , w.l.o.g., P uv = ( u, p 1 , . . . , p r , v ). Note th at u 6 = v , else C = ( u, p 1 , . . . , p r ) and the ﬁrst case of Rule 2 would ha v e applied. If r = 1 and P uv = ( u, p 1 , v ), then by the marking pro cess of Rule 2 and as K wa s n ot marked, there must b e ℓ + 1 cliques whic h con tain a sh ared neigh b or of u and v (whic h were marke d and ) whic h are present in G ′ . Hence, by Ob serv ation 2, at least one of those cliques, K ′ sa y , was n ot used b y C , and we ma y replace p 1 b y any s hared neigh b or of u and v in K ′ . W e obtain a cycle C ′ that has one fewer subpath whic h is not in G ′ − X con tradicting our c hoice of C . No w, if r > 1, then P uv = ( u, p 1 , . . . , p r , v ) with p 1 6 = p r . Hence, K con tains a neigh b or of u and a diﬀeren t neigh b or of v . As K is not mark ed, the redu cti on rule must h a v e marke d ℓ + 1 other cliques whic h are at le ast a s la rge as K , and which conta in suc h neighbors. Again, b y Obs er v ation 2, there must b e a clique K ′ among them which is not used by C . Th e clique K ′ is at least as large as K and it con tains vertice s p and q with p adj ac ent to u and q adjacen t to v . Hence, we ma y replace the su bpath P uv of C by a path fr om u to p , follo w ed b y a p ath from p to q using all vertice s of K ′ , and bac k to v ; w e call this P ′ uv . Clearly , P ′ uv is at least as long as P uv , since P uv could at most use all v ertices of K (there might b e other su bpaths of C usin g K ) and K ′ has at least that man y v ertices. Thus, replacing P uv in C by P ′ uv w e obtain a cycle C ′ of at least th e s ame length, whic h u ses one few er su b path that is not con tained in G ′ − X , con tradicting the c hoice of C . It follo ws that C con tains on ly ve r tices of X and of mark ed cliques. Hence, it exists also in G ′ , completing the pr o of. F or the r emaining r eduction we pr oceed sim ilarly to Section 4.2: First, we show that we can without harm restrict to allo wing only a small num b er of v ertices in eac h clique f or connecting to X , the r emainin g v ertices are only needed to p ossibly extend the length of the cycle (i.e ., they are visited after en tering the clique and b efore lea ving it). This allo ws for a s tr ai ghtforw ard FPT-algorithm of ru n time O ( ℓ 10 ℓ · n c ). Con s equen tly , w e may assum e the num b er of vertic es to b e b ound ed b y O ( ℓ 10 ℓ ), else solving the w hole instance in time O ( n c +1 ). Thus, the num b er of additional vertice s in eac h clique may b e enco ded in b inary , with co ding length O ( ℓ log ℓ ), giving a p olynomial sized equiv ale nt instance of a sp ecial v ersion of our pr oblem. Finally , w e use a Karp reduction from this sp ecial ve rs io n bac k to the basic p roblem, to obtain a p olynomial k ernel. Lemma 5. Given an instanc e ( G, k , X ) one c an in p olynom ial time identify ( ℓ + 1 ) 3 vertic es in e ach cliqu e of G − X , such that if ( G, k , X ) is y es , then G c ontains a cycle of length at le ast k that enters and le aves cliques of G − X only thr ough the identiﬁe d vertic es. Pr o of. Giv en ( G, k , X ) mark f or eac h pair of v ertices u, v ∈ X up to 2 ℓ + 1 sh ared n eig hb ors in eac h clique. F urthermore, w e mark for eac h ve r tex v ∈ X up to 2 ℓ + 1 neigh b ors in eac h clique. 11 No w, let us assu me that G conta ins a cycle of length at least k , and let C b e a cycle of length at least k that enters and lea ves cliques of G − X through a minim um num b er of un m ark ed vertices (equiv ale ntly , C uses a minimum n u m b er of ed ges b et w een the mo dulator and unmarked v ertices). Assume th at C = ( . . . , u, p 1 , . . . , p r , v , . . . ) where u, v ∈ X , all vertice s p i are from the same clique K of G − X , and at least p 1 is un m ark ed. If r = 1, then C = ( . . . , u, p 1 , v , . . . ). As p 1 is un mark ed, the clique K m ust conta in 2 ℓ + 1 mark ed v ertices that are shared n ei ghb ors of u and v , sa y q 1 , . . . , q 2 ℓ +1 . If at least one of them is not used by C , sa y q i , th en we ma y replace p 1 b y q i , an d ther eby obtain a cycle of the same length as C that uses one less unmarked vertex to enter and lea ve cliques; a contradictio n to th e c hoice of C . So assume that C uses all v ertices q 1 , . . . , q 2 ℓ +1 . By ℓ = | X | we know that at least one v ertex, say q j , is not adjacen t to v ertices of X on C , b ut m us t b e adjacent to other vertices of K . Therefore, w e may swap the p ositions of p 1 and q j in C : I ndeed, q j is adjacen t to u and v , and p 1 is adjacen t to all other vertices of K (and clearly it is not adjacen t to q j on C ). W e obtain a cycle of the same length, whic h uses on e less u nmark ed vertex to en ter or lea v e cliques of G − X , con tradicting our c hoice of C : The new cycle no w enters a clique at q j instead of p 1 . No w, w e consider the case that r > 1, so C = ( . . . , u, p 1 , . . . , p r , v , . . . ) w ith p 1 6 = p r . Since p 1 is unmarked, there must b e 2 ℓ + 1 mark ed v ertices in K which are adjacen t to u , say q 1 , . . . , q 2 ℓ +1 . It follo ws that at least one of those vertices, sa y q j , is not adjacen t to X on C . W e can no w apply the same switc h in g argument s as in th e previous p aragraph to obtain a cycle C ′ of the same length, whic h use one less u nmark ed ve rtex to ente r or lea ve cliques of G − X , con tradicting our c h oi ce of C . It f ol lows that C en ters and lea v es cliques only through marked vertic es. The mark ed ve r tices constitute the sets of v ertices in the cliques as claimed by the lemma. Th ere are at m ost ℓ · (2 ℓ + 1) +  ℓ 2  · (2 ℓ + 1) ≤ ( ℓ + 1) 3 mark ed vertic es p er clique of G − X , as claimed. Lemma 6. Given ( G, k , X ) , with ℓ := | X | , a cycle of length at le ast k in G c an b e found in time O ( ℓ 10 ℓ · n c ) = 2 O ( ℓ log ℓ ) n O (1) if it exists. Pr o of. W e use Reduction Rule 2 to redu ce the n u m b er of cliques in G − X to O ( ℓ 3 ). Then w e mark v ertices according to Lemma 5. A ss uming that ( G, k , X ) is a yes -in s ta n ce , it follo ws that there m ust b e a cycle of length at least k in G which ente rs and lea ves cliques only in marked vertice s. The follo wing brute force algorithm ﬁnds suc h a cycle if it exists: 1. If any clique has size at least k then retur n yes . E lse at least one v ertex of X must b e used. 2. T ry all ordered subsets X ′ = { x 1 , . . . , x t } of X for the in tersection of C with X . 3. F or all pairs ( x i , x i +1 ) (including ( x t , x 1 )) try all cliques that contai n neigh b ors of x i and x i +1 , or, if p ossible, try the edge { x i , x i +1 } to connect x i and x i +1 on C . 4. If a clique K h as b een c hosen to connect x i and x i +1 on C , then try all marked v ertices of the clique for en tering and lea ving K on C , i.e., to ﬁ nd either ve r tic es p and q with C = ( . . . , x i , p, . . . , q , x i +1 ), or a single verte x p for C = ( . . . , x i , p, x i +1 , . . . ) (and chec k adjacency of p and p ossibly q to x i and x i +1 ). 5. Greedily connect the c hosen v ertices ins ide the cliques: F or eac h clique where we ha v e chosen the option C = ( . . . , x i , p, . . . , q , x i +1 , . . . ) we can include all remainin g ve r tic es of the clique in to our cycle. I f only s hared neigh b ors w ere chosen, then this is not p ossible. 12 It is easy to see th at giv en the existence of any cycle C ′ of length at least k w hic h ent ers and lea v es cliques only th rough mark ed vertice s, our simp le algo r ith m must ﬁ nd a cycle of at least the same length: It will even tually try the same c hoice and order of v ertices from X , and pick the same marked vertic es to conn ec t th em to cliques. At that p oin t, it is easy to see, that the greedy connection of the chosen marke d vertic es must giv e a cycle of at least the same length. C lea rly , if our algorithm ﬁnd s a cycle of length at least k , then the instance is yes . W e close b y giving an upp er boun d on the dep endence of ℓ in the run time. It is easy to see that the dep endence on the inp ut size is (a lo w d eg ree) p olynomial. 1. Th is s te p can b e p erformed in p olynomial time. 2. Th ere are less than ( ℓ + 1) ℓ = O ( ℓ ℓ ) ordered su bsets X ′ of X . 3. W e pic k up to ℓ cliques (p ossibly with rep etition) out of th e O ( ℓ 3 ) cliques of G − X , for a total of O ( ℓ 3 ℓ ) c hoices. 4. Up to ℓ times, we pic k t wo marked v ertices among th e ( ℓ + 1) 3 v ertices, i.e., we ha v e O ( ℓ 6 ℓ ) c hoices. 5. Th is s te p can b e p erformed in time p olynomial in the inpu t size. This giv es a total runtime of O ( ℓ 10 ℓ · n c ) = 2 O ( ℓ log ℓ ) n O (1) . Theorem 6. Long Cycle p arameterized b y a modula tor to cluste r graphs admits a p olynomial kernel. Pr o of. Let ( G, k , X ) b e an in put in s ta n ce . W.l.o.g. w e let the instance b e reduced according to Reduction Ru le 2 , i.e., G has at most O ( ℓ 3 ) cliques. If G has at least 2 O ( ℓ log ℓ ) v ertices, then the algorithm of Lemma 6 can b e used to solv e the instance in time n · n O (1) = n O (1) . Oth er w ise, using n < 2 O ( ℓ log ℓ ) w e enco de our instance in the follo wing wa y , as an instance of a diﬀerent problem: • W e mark vertices according to Lemma 5. • In eac h clique, w e replace all unmarked v ertices b y a single unmarke d verte x with an in teger lab el stating the n umber of unmark ed vertice s. In binary encodin g, that lab el n ee d s at most log n = log (2 O ( ℓ log ℓ ) ) = O ( ℓ log ℓ ) bits. F or th e so-encod ed instance we ask f or a path of length at least k whic h en ters and lea ves cliques only through marked vertice s, b ut w e accoun t the lab eled v ertices as su bpaths with a num b er of v ertices equal to their lab el. C lea rly , this is equiv alen t to the original question. In addition to the v ertices of X , the instance has at most O ( ℓ 3 ) cliques eac h with at most ( ℓ + 1) 3 + 1 v ertices plus one inte ger lab el of bit size at m ost O ( ℓ log ℓ ) p er clique. Clearly , this giv es a total size which is p olynomial in ℓ . Finally , w e observ e that the question wh ic h we ask ab out this instance can b e answ ered in NP . As Long Cycle is NP-complete, it remains so even when w e app end the d istance to a cluster graph in unary as wel l as a reasonable enco ding of a mo d ulator X as b oth are b ou n ded by n . Th u s there is a Karp reduction from our alternativ e problem to Lon g Cycle p arameterized by a modula tor to cluster graphs . By standard argumen ts (e. g. see [7]) th is im p lies that the latter problem admits a p olynomial kernel. 13 F urther problems parameterized by a mo dulator to cluster graphs. F or Long P a th , Hamil t onian Cycle , and Hamil tonian P a t h p olynomial k ernels follo w directly fr om Theorem 6 via straigh tforwa rd r eductio n s to the Lon g Cycl e problem. Ho wev er, it can b e easily observed that for Hamil tonian Cycle and Hamil tonian P a t h th e enco ding tric k is not necessary . In deed all the unmarked v ertices can alwa ys b e assumed to o ccur in a single subpath of the ﬁn al Hamiltonian cycle or path. Hence, it su ﬃces to ke ep only a sin gl e unmarked verte x p er clique. T h is sa ves the argumen t via Karp reductions. In f act, the m arking argument can b e tight ened to o, b y using the matc hing app roac h from Section 4.1. Corollary 3. Long P a t h , Hamil tonian Cycle , and Hamil tonian P a th p ar ameterize d by a mo dulator to cluster gr aphs admit p olyno mial k ernels. Pr o of. F or an instance ( G , k , X ) of Lo ng P a th paramete rized b y a mo dulator fr om clus te r graphs, simply add a universal vertex v adjacen t to all vertic es of G and ask for a cycle of length k + 1. F u r thermore, the verte x v is added to the mo dulator X , increasing the parameter v alue b y one. Theorem 6 no w creates an intermediate in stance of size O ( | X | 6 ) w hic h reduces bac k to Long P a th b y a Karp reduction. (Note that we ma y also remov e the universal v ertex to get an instance of Long P a th with lab eled v ertices. Of course th is is a compression, or generalized kernelizat ion.) It is straigh tforward to get similar reductions for Hamil tonian Cycle and Hamil tonian P a th b y asking for long cycles or path resp ectiv ely of length (at least) equal to the total n u m b er of vertic es. Ho w eve r , it can b e easily seen that follo wing the p roof of Theorem 6 for an obtained instance of Long Cycle , one may r ep la ce e ach lab eled vertex b y a single unlab eled one and up date the target length (suc h that it alwa ys matc h es the num b er of v ertices). In deed, any cycle u s ing all v ertices can b e mo diﬁed to conta in all unm arked vertic es as a sin gle su bpath. The n umb er of those vertice s (i.e. the single lab el in eac h clique) is then immaterial. Thus one obtains graphs with O ( | X | 6 ) v ertices. I t is straigh tforw ard to make the mo diﬁcations to get bac k to instances of Hamil t onian Cyc le and Hamil to nian P a th resp ectiv ely . W e n ow turn our atten tion to Disjoint P a ths and Disjoint Cycles . I n b oth problems th e length of paths or cycles is not important, but there ma yb e large n umb ers of paths and cycles inside eac h clique. Observ at ion 3. In the Disjoint P a ths problem parameterized b y a mo dulator X to a cluster graph w e ma y assu me that all r equested pairs lie en tirely in X : First, it is clear that requ est pairs ( s i , t i ) con tained in the same clique of G − X , hav e the un iquely optimal p ath consisting only of s i and t i . Th us, all su c h vertex pairs ma y b e deleted (b oth from the graph and the list of requests). T o satisfy any other request pair the corresp onding path needs to int ersect X , b oundin g the m axim um feasible n umb er of requests b y | X | ; else we reject th e instance. Thus, including the v ertices of all requests at most triples the size of X . F rom th is obs erv ation it is clear th at a p olynomial kernelizat ion for Disjoint P a t hs can b e ac hiev ed by marking (or matc hing) enough v ertices to allo w for the necessary paths. The case for Disjoint Cycle s is similar: Ther e are at most | X | cycles which include v ertices of X . All furth er cycles can b e c hosen as triangles inside single cliques. Again a marking pr ocedur e su ﬃces, adding an add itional step of r emoving triples of unmarked v ertices f rom an y sin gle clique and eac h time reducing the target n umb er of cycles b y one (accoun ting for those trivial cyc les not in tersecting X ). Corollary 4. Disjoint P a ths and Disjo int Cycles p ar ameterize d by a mo dula tor to a cluster gr aph admit p olynomial kernels. 14 5 Lo w er b ounds for path and c ycle pr oble ms In this section w e present k ernelization lo we r b ounds for the directed- an d und irecte d v arian ts of Hamil t onian Cycle with s tructural parameters. The parameterizations w e us e are at least as large as the treewidth of the inpu t graph s (or the un derlying undir ected graph, in the directed case) which sho ws that the parameterized problems for wh ic h we p ro v e a k ern el lo wer b oun d are indeed contai n ed in FPT. W e ﬁrs t develo p a lo wer b ound for Direct ed Ha mil tonian Cycle p ar ameterized by a modul a tor to Bi-p a ths , and the show h o w to alter this construction to giv e lo wer b ound for Hamil t onian Cycle p a rameterized by a modula tor to Outerpla nar graphs . Ou r pr oofs us e the tec hniqu e of cross-comp osition [6], in whic h a k ernel lo wer b ound is obtained by sho win g th at the logical O R of a series of ins tances of an NP-hard p roblem, can b e em b edded in a sin gle instance of the parameterized target problem at a small p aramete r cost. W e therefore start eac h subsection b y deﬁning an appropriate NP-h ard pr oblem to comp ose, and then giv e a cross-comp ositio n algorithm. 5.1 Directed Hamiltonian Cyc le with a mo dulator to bi-paths W e start b y deﬁning the NP-hard pr oblem wh ic h w e w ill use in the cross-comp osition. Hamil t onian s − t P a th in Directe d Bip ar tite Graphs Input: A bip artit e digraph D with color classes A = { a 1 , . . . , a n A } and B = { b 1 , . . . , b n B } with n B = n A + 1 su c h that N − D ( b 1 ) = ∅ and N + D ( b n B ) = ∅ . Question: Do es D conta in a directed Hamiltonian path which starts in b 1 and ends in b n B ? It is not d iﬃcult to s ho w that this p roblem is NP-complete. Prop osition 1. Ha mil tonian s − t P a th in Directed Bip ar tite Graphs is NP - c omplete. Pr o of. Mem b ership in NP is trivial. W e pro ve hardn ess by a r eduction from Hamil tonian s − t P a th whic h is a classical NP-complete problem [22, GT39]. On input an und irecte d graph G with distinguished vertices s, t construct a graph G ′ on v ertex set V ( G ) × { 1 , 2 , 3 , 4 } with edges {{ v 1 , v 2 } , { v 2 , v 3 } , { v 3 , v 4 } | v ∈ V ( G ) } ∪ {{ v 1 , u 4 } , { v 4 , u 1 } | { u, v } ∈ E ( G ) } . It is easy to verify that G has a Hamiltonian s − t path if and only if G ′ has a Hamiltonian s 1 − t 4 path. T he graph G ′ is bipartite. T o obtain an instance of Hamil tonian s − t P a th in Directed Bip ar tite Graphs we bu ild a digraph D ′ b y replacing th e edges in G ′ with bidir ectional arcs, addin g a new starting vertex s ∗ with un ique out-neighb or s 1 , addin g a v ertex w w ith in - neigh b or t 4 and a new ending v ertex t ∗ with in-neigh b or w . T aking the appropr ia te partite sets of the b ipartitio n , lab eling s ∗ as b 1 and t ∗ as b n B w e obtain an equiv alen t instance of Ham il tonian s − t P a th in Direct ed Bip ar tite Graphs . No w we formally deﬁne the parameterized pr oblem for whic h we w ill prov e a kernel lo we r b ound. Directed Hamil tonian Cycle p a rameterized b y a mo dula tor to Bi-p a ths Input: A digraph D and a mo dulator X ⊆ V ( D ) such th at D − X ∈ Bi-p a ths . P arameter: The size | X | of the mo dulator. Question: Do es D ha v e a directed Hamiltonian cycle? 15 The follo wing prop ositions will b e useful for v erifyin g the correctness of the comp osition. Prop osition 2. L et D b e a digr aph with dir e c te d Hamiltonian cycle C . If v ∈ V ( D ) is a vertex such that N D ( v ) = { u, w } then C c ontains either the p ath ( u, v ) , ( v , w ) or ( w , v ) , ( v , u ) . Prop osition 3. L et D b e a bip artite digr aph with c olor classes A = { a 1 , . . . , a n A } and B = { b 1 , . . . , b n B } with n B = n A + 1 . If C ⊆ A ( D ) is a set of ar cs such that: 1. D [ C ] do es not c ontain a dir e cte d cycle, 2. al l ve rtic es a ∈ A ar e the he ad of one ar c in C and the tail of one ar c in C , 3. no vertex b ∈ B is the he ad of two ar cs in C , or the tail of two ar cs in C , 4. the vertex b 1 is the he ad of one ar c in C and the tail of no ar c, 5. the vertex b n B is the tail of one ar c in C and the he ad of no ar c, then C is a Hamiltonian p ath fr om b 1 to b n B in D . W e can no w giv e the cross-comp osit ion. Theorem 7. Directed Hamil tonian Cycle p ar ameterized b y a modula tor t o Bi-p a ths do es not admit a p olynom ial kernel unless NP ⊆ c oNP / p oly. Pr o of. By Th eo r em 1 and Pr oposition 1 it is su ﬃcie nt to sh ow that Ham il tonian s − t P a th in Directed Bip ar tite Graphs cross-comp oses into Directe d Hamil tonian Cycle p arame- terized by a modul a tor to Bi-p a ths . W e ﬁrst giv e th e p olynomial equiv alence relatio ns hip R to b e used for the cr oss-comp osition. Fix some r ea sonable enco ding of instances of Ham il tonian s − t P a th in Direct ed Bip ar tite Grap hs in to an alph abet Σ suc h that w ell-formed instances can b e recognized in p olynomial time. No w say that t w o strings in Σ ∗ are equiv alen t under R if (a) they are b oth malformed in sta n ce s, or (b) they enco de instances ( D 1 , A 1 , B 1 ) and ( D 2 , A 2 , B 2 ) of Hamil t onian s − t P a th in Direct ed Bip ar tite Graphs suc h that | A 1 | = | A 2 | and | B 1 | = | B 2 | . It is not hard to v erify that a set o f strings whic h enco de instances o f up to n v ertices eac h, is parti- tioned into O ( n ) equiv alence classes by R , wh ic h is therefore a p olynomial equiv alence r ela tionship . It no w suﬃces to giv e an algorithm whic h comp oses a sequence of instances of Ha mil tonian s − t P a th in Directed Bip a r tite Gra phs whic h are equiv alen t under R in to one instance of Directed Hamil tonian Cycle p a rameterized by a modul a tor to Bi-p a ths . I f th e input consists of malformed in sta n ce s then w e can simp ly output a constan t-size no -ins ta n ce . Hence in the remainder we may assume th at the input cont ains r well -formed in stance s ( D 1 , A 1 , B 1 ) , . . . , ( D r , A r , B r ), and th at | A i | = n A and | B i | = n B for i ∈ [ r ] with n B = n A + 1. Lab el the ve rtices in eac h set A i as a i, 1 , . . . , a i,n A and the v ertices of a s et B i as b i, 1 , . . . , b i,n B for i ∈ [ r ]. Recall that instance i asks wh ether D i has a Hamiltonian path fr om b i, 1 to b i,n B . W e construct a digraph D ∗ as follo ws. 1. F or i ∈ [ r ], for j ∈ [ n A ] add v ertices a ′ i,j , a ′′ i,j , a ′′′ i,j to D ∗ , and add arcs ( a ′ i,j , a ′′ i,j ) , ( a ′′ i,j , a ′ i,j ) , ( a ′′ i,j , a ′′′ i,j ) , ( a ′′′ i,j , a ′′ i,j ) . 2. As the n ext step w e add one-dir ectional arcs to connect adjacen t triples. F or i ∈ [ r ], f or j ∈ [ n A − 1] add the arc ( a ′′′ i,j , a ′ i,j +1 ). 16 a 1 a 2 a 3 a 4 b 1 b 2 b 3 b 4 b 5 (a) Input instance ( D 1 , A 1 , B 1 ) of Hamil tonian s − t P a th i n Di - rected Bip ar tite Gra phs . x 1 a ′ 1 , 1 a ′′ 1 , 1 a ′′′ 1 , 1 a ′ 1 , 2 a ′′ 1 , 2 a ′′′ 1 , 2 a ′ 1 , 3 a ′′ 1 , 3 a ′′′ 1 , 3 a ′ 1 , 4 a ′′ 1 , 4 a ′′′ 1 , 4 y 1 x 2 a ′ 2 , 1 a ′′ 2 , 1 a ′′′ 2 , 1 a ′ 2 , 2 a ′′ 2 , 2 a ′′′ 2 , 2 a ′ 2 , 3 a ′′ 2 , 3 a ′′′ 2 , 3 a ′ 2 , 4 a ′′ 2 , 4 a ′′′ 2 , 4 y 2 z b ∗ 1 b ∗ 2 b ∗ 3 b ∗ 4 b ∗ 5 (b) Output instance of Directed H amil tonian Cycle p arameterized by a modula tor to Bi -p a ths . Figure 1: An example of the lo wer-boun d construction of Th eorem 7 when comp osing r = 2 inputs with n A = 4 and n B = 5. (a) The ﬁrs t input instance. (b) Resulting outp u t instance. The arcs b et w een { b ∗ 1 , . . . , b ∗ 5 } and { a ′ 2 ,j , a ′′ 2 ,j , a ′′′ 2 ,j | j ∈ [4 ] } which enco de the second input ( D 2 , A 2 , B 2 ) ha ve b een omitted for readabilit y . 3. F or eac h in s ta n ce i ∈ [ r ] add t wo s p ecial vertice s x i and y i , tog ether w ith arcs ( x i , a ′ i, 1 ) and ( a ′′′ i,n A , y i ). F or i ∈ [ r − 1] add the arcs ( y i , x i +1 ). 4. Ob serv e that at this stage, D ∗ ∈ Bi-p a ths . All ve rtices we add from this p oint on will go in to the mo dulator X ∗ suc h th at D ∗ − X ∗ will b e a mem b er of Bi-p a ths . 5. W e add a sp ecial v ertex z with arcs ( x i , z ) an d ( z , y i ) for i ∈ [ r ]. 6. F or j ∈ [ n B ] add a ve r te x b ∗ j to the graph D ∗ , and let B ∗ b e the set of these vertic es. Add arcs ( y r , b ∗ 1 ) and ( b ∗ n B , x 1 ). 7. As the last step of the construction w e re-enco de the b ehavio r of the input grap h s D i in to the instance. F or i ∈ [ r ], for all arcs ( a i,j , b i,h ) in A ( D i ) add th e arc ( a ′ i,j , b ∗ h ) to D ∗ . F or all arcs ( b i,j , a i,h ) ∈ A ( D i ) add ( b ∗ j , a ′′′ i,h ) to D ∗ . This concludes the d escrip tio n of D ∗ , whic h is illustrated in Fig. 1. No w deﬁn e X ∗ := { z } ∪ B ∗ . The output of the cross-comp osition is th e instance ( D ∗ , X ∗ ) of Directed Hamil tonian Cycle p arameterize d by a modula tor to Bi-p a ths . It is easy to v erify that D ∗ − X ∗ ∈ Bi-p a ths , and that the construction can b e carried out in p olynomial time. The parameter | X ∗ | is b ounded b y 1 + n B whic h is suﬃcient ly small. It remains to p ro v e that D ∗ 17 is yes if and only if one of the input instances is y es . Before provi n g this equiv ale n ce we establish some prop erties of D ∗ . Claim 1. L et C ⊆ A ( D ∗ ) b e a dir e cte d Hamiltonian cycle in D ∗ . The fol lowing must hold. 1. If ( a ′′′ i,j , a ′′ i,j ) is an ar c on C for some i ∈ [ r ] , j ∈ [ n A ] then ther e ar e distinct i nd ic es f , f ′ such that vertic es b ∗ f , a ′′′ i,j , a ′′ i,j , a ′ i,j , b ∗ f ′ app e ar c onse cutively on C . 2. If ( x i , a ′ i, 1 ) is not an ar c on C for some i ∈ [ r ] , then none of the ar cs ( a ′′′ i,j , a ′ i,j +1 ) for j ∈ [ n A − 1] ar e c ontaine d in C , nor is the ar c ( a ′′′ i,n A , y i ) . Pr o of. W e prov e the diﬀerent comp on ents consecutiv ely . 1. Assu me that ( a ′′′ i,j , a ′′ i,j ) is an arc on C . By construction of D ∗ the in -neig hb ors of a ′′′ i,j whic h are not a ′′ i,j are of the form b ∗ f for f ∈ [ n B ], and since a ′′ i,j is used as the successor of a ′′′ i,j this sho ws that the p redecessor of a ′′′ i,j m ust b e some b ∗ f . Since N D ∗ ( a ′′ i,j ) = { a ′ i,j , a ′′′ i,j } w e ﬁnd by Prop osition 2 that the vertic es a ′′′ i,j , a ′′ i,j , a ′ i,j m ust b e consecutiv e on C . S imila r ly as b efore, the construction of D ∗ sho ws that the only out-neigh b ors of a ′ i,j diﬀeren t than a ′′ i,j are of the form b ∗ f ′ for f ′ ∈ [ n B ]. Since a ′′ i,j is used as predecessor of a ′ i,j on C , it cannot b e the successor and hence some b ∗ f ′ m ust b e the successor of a ′ i,j on C whic h sho w s that b ∗ f , a ′′′ i,j , a ′′ i,j , a ′ i,j , b ∗ f ′ are consecutiv e on C . 2. Assu me that ( x i , a ′ i, 1 ) is not on C , but at least one of the arcs in the set Z i := { ( a ′′′ i,j , a ′ i,j +1 ) | j ∈ [ n A − 1] } ∪ { ( a ′′′ i,n A , y i ) } is used on C . Let j ∗ b e smallest in dex s uc h that the v ertex a ′′′ i,j ∗ is the head of an arc in C ∩ Z i . T hen a ′′ i,j ∗ is not the successor of a ′′′ i,j ∗ on C and by Prop osition 2 it m ust therefore b e its predecessor, sho wing that a ′ i,j ∗ , a ′′ i,j ∗ , a ′′′ i,j ∗ m ust b e consecutiv e on C . If j ∗ ≥ 2 then the only in-neigh b ors of a ′ i,j ∗ in D ∗ are { a ′′′ i,j ∗ − 1 , a ′′ i,j ∗ } , and if j ∗ = 1 then the only in-neighb ors are { x i , a ′′ i,j ∗ } . By c h oice of j ∗ as the smal lest index fr om Z i whic h is the head of an arc in C ∩ Z i , the ﬁrst in-neigh b or of a ′ i,j ∗ cannot b e its predecessor on C . But since a ′′ i,j ∗ is its successor on C , that v ertex cannot b e its p redecessor either. So a ′ i,j ∗ do es not ha ve a predecessor on C whic h con tradicts the assumption that C is a Hamilt onian cycle, whic h conclud es the pro of of this part. W e are no w ready to pro ve that ( D ∗ , X ∗ ) is yes if and only if on e of the input ins ta n ce s is yes . F or the ﬁrst dir ec tion, assum e that D ∗ has a d irecte d Hamiltonian cycle C . Sin ce N − D ∗ ( z ) = { x i | i ∈ [ r ] } there is an index i ∗ suc h that x i ∗ is the pred ecessor of z on C , which sho ws that the arc ( x i ∗ , a ′ i ∗ , 1 ) is not used on C . By (2 ) this implies th at there are n o arcs ( a ′′′ i ∗ ,j , a ′ i ∗ ,j +1 ) for j ∈ [ n A − 1] con tained in C , nor is the arc ( a ′′′ i ∗ ,n A , y i ∗ ). By construction of D ∗ this imp lies that eac h vertex a ′′′ i ∗ ,j for j ∈ [ n A ] has a ′′ i ∗ ,j as its s uccesso r on C (there is only one other op tion for the successor, whic h is explicitly excluded). Hence all the arcs ( a ′′′ i ∗ ,j , a ′′ i ∗ ,j ) are con tained in C for j ∈ [ n A ]. By (1) this sho ws th at for eac h j ∈ [ n B ] there are b ∗ f , b ∗ f ′ suc h that b ∗ f , a ′′′ i ∗ ,j , a ′′ i ∗ ,j , a ′ i ∗ ,j , b ∗ f ′ are consecutiv e on C . No w consider the set of arcs C i ∗ in D i ∗ deﬁned as follo ws: • If b ∗ f , a ′′′ i ∗ ,j , a ′′ i ∗ ,j , a ′ i ∗ ,j , b ∗ f ′ are consecutive on C then add the arcs ( b i ∗ ,f , a i ∗ ,j ) and ( a i ∗ ,j , b i ∗ ,f ′ ) to C i ∗ . Recall that b y construction of D ∗ , the arc ( b ∗ f , a ′′′ i ∗ ,j ) is on ly present in D ∗ when ( b i ∗ ,f , a i ∗ ,j ) ∈ A ( D i ∗ ), and the arc ( a ′ i ∗ ,j , b ∗ f ) is only p resen t in D ∗ when ( a i ∗ ,j , b i ∗ ,f ) ∈ A ( D i ∗ ), and hence all arcs added to C i ∗ b y th is deﬁnition are indeed present in D i ∗ . W e w ill sho w that the set C i ∗ ⊆ A ( D i ∗ ) satisﬁes 18 all requiremen ts of Prop osition 3 for graph D i ∗ , thereb y sho w in g that D i ∗ has a Hamilto nian path from b i ∗ , 1 to b i ∗ ,n B . If C i ∗ con tains a directed cycle, then by construction of C i ∗ it follo ws that C must con tain a closed cycle on a verte x su bset of { b ∗ j | j ∈ [ n B ] } ∪ { a ′ i ∗ ,j , a ′′ i ∗ ,j , a ′′′ i ∗ ,j | j ∈ [ n A ] } sho w in g that C is not a Hamiltonian cycle in D ∗ ; hence the set C i ∗ satisﬁes (1) of Prop osition 3. The deﬁn iti on of C i ∗ directly sh o ws that C satisﬁes (2). If some v ertex b i ∗ ,j is the head or tail of tw o arcs in C i ∗ then th e corresp onding vertex b ∗ j is head or tail of t wo arcs in the Hamiltonian cycle C , wh ic h is not p ossib le; hence (3) is s atisﬁed. Since the deﬁnition of Hamil tonian s − t P a th in Directe d Bip ar tite Graphs guarantee s that b i ∗ , 1 do es not ha ve in -arcs in D i ∗ , and that b i ∗ ,n B has no out-arcs in D i ∗ , th e vertex b ∗ 1 cannot o ccur as s uccesso r to a verte x a ′ i ∗ ,j and vertex b ∗ n B cannot o ccur as pr ed ec essor of a ve r te x a ′′′ i ∗ ,j . Th erefore C i ∗ con tains n o arcs leading into b i ∗ , 1 and no arcs leading out of b i ∗ ,n B , pro ving that the last co nd itio n is also satisﬁed. By the p rop osit ion this pro ve s that C i ∗ is a Hamiltonian b i ∗ , 1 − b i ∗ ,n B path in D i ∗ whic h p ro v es this d irect ion of the equ iv alence. The other direction is straigh t-forward. Assu me that C i ∗ is a Hamiltonian path fr om b i ∗ , 1 to b i ∗ ,n B in D i ∗ . W e construct a Hamiltonian cycle C in D ∗ as follo ws. • F or i 6 = i ∗ add all arcs b et we en consecutiv e v ertices of x i , a ′ i, 1 , a ′′ i, 1 , a ′′′ i, 1 , a ′ i, 2 , . . . , a ′′′ i,n A − 1 , a ′ i,n A , a ′′ i,n A , a ′′′ i,n A , y i to C . • Add all arcs ( a ′′′ i ∗ ,j , a ′′ i ∗ ,j ) , ( a ′′ i ∗ ,j , a ′ i ∗ ,j ) for j ∈ [ n A ] to C . • Add all arcs ( y i , x i +1 ) for i ∈ [ r − 1] to C . • Add th e arcs ( b ∗ n B , x 1 ) and ( y r , b ∗ 1 ) to C . • F or eac h arc ( b i ∗ ,f , a i ∗ ,j ) ∈ C i ∗ add ( b ∗ f , a ′′′ i ∗ ,j ) to C . • F or eac h arc ( a i ∗ ,j , b i ∗ ,f ) ∈ C i ∗ add ( a ′ i ∗ ,j , b ∗ f ) to C . Using the construction of D ∗ is it straigh t-forwa r d to verify th at C is a Hamiltonian cycle in D ∗ , whic h p ro v es the rev erse d irecti on of the equiv alence and concludes th e pro of. It is not diﬃcult to see th at the pro of of T heorem 7 can b e adap ted to giv e a k ern el lo wer boun d for the v arian t wh ere w e are lo oking for a Hamiltonian path instead of a Hamiltonian cycle; these b ounds in turn imply that the versions wh ere w e are lo oking for a long path or cycle (instead of one w h ic h is Hamiltonian) are at least as hard to kernelize, as is the case w hen w e wa nt to ﬁnd a long s − t path or a long cycle through a giv en ve rtex. 5.2 Hamiltonian Cycle with a mo dulator to outer pla nar graphs F or the cross-comp ositio n of this section we will u se the follo wing v arian t of Ham i l tonian P a th on undirected bip artit e graphs. Hamil t onian s − t P a th in Bip ar tite Graphs Input: A bipartite graph G with co lor classes A = { a 1 , . . . , a n A } and B = { b 1 , . . . , b n B } with n B = n A + 1 su c h that | N G ( b 1 ) | = | N G ( b n B ) | = 1. Question: Do es G con tain a Hamiltonian p ath whic h starts in b 1 and ends in b n B ? NP-completeness of this problem follo ws from the construction of Pr oposition 1. 19 a − ˆ a + ˆ a − a + (a) Domino. a − ˆ a + ˆ a − a + (b) a -trav ersal. a − ˆ a + ˆ a − a + (c) ˆ a -trav ersal. Figure 2: (a) Th e domino gadget with four terminal vertice s a − , a + , ˆ a − , ˆ a + . If graph G cont ains the domino as a subgraph suc h th at only these term in als are connected to the remainder of the graph, then an y Hamiltonian cycle for G must use the a -tra versal (b) or ˆ a -tra ve r sal (c) of the d omino. Prop osition 4. The Hamil t onian s − t P a t h in Bip ar tite Graphs pr oblem is NP- c omplete. The problem for w h ic h w e p ro v e a low er b ound is formally d eﬁned as follo ws. Hamil t onian Cycle p arameterized by a modul a tor to Oute rplanar graphs Input: A graph G and a mo dulator X ⊆ V ( G ) such that G − X ∈ Outerplanar . P arameter: The size | X | of the mo dulator. Question: Do es G ha v e a Hamiltonian cycle? W e can mo dify the construction of Theorem 7 to giv e a low er b ound for Hamil tonian Cycle p ar ameterized by a modula tor to Oute rplanar gra phs . O ur main to ol is the “domino” gadget of Fig. 2, which a Hamiltonian cycle must visit in one of tw o sp eciﬁc wa ys. Th is domino will b e used to sim u lat e directed edges. Prop osition 5. L et G b e an undir e cte d gr aph c onta ining the domino as an induc e d sub g r aph, such that only the vertic es lab ele d a + , a − , ˆ a + , ˆ a − have neighb ors outside the domino. Then any Hamil- tonian cycle in G must either (a) c ontain an a -tra v ersal of the domino, which i s p ath b etwe en a + and a − , v isiting al l other vertic es of the domino in b etwe en or (b) c ontain an ˆ a -trav ersal of the domino, which is a p ath b etwe en ˆ a + and ˆ a − , visiting al l other vertic es in b etwe e n. Theorem 8. Hamil tonian Cycle p aramete rized b y a mo dula tor to Ou terplanar grap hs admits no p olynomial kernel unless NP ⊆ c oNP / p oly. Pr o of. By Theorem 1 and Prop osition 4 it is suﬃcient to sh o w th at H amil tonian s − t P a th in Bip ar tite Graph s cross-comp oses into Hamil tonian Cycle p arameterized by a mod- ula tor to Outerplana r graphs . By arguments similar to that in the p r oof of Theorem 7 w e can deﬁne a suitable p olynomial equiv alence relationship R such that it is su ﬃcien t to giv e an algorithm w hic h, give n r wel l-formed instances ( G 1 , A 1 , B 1 ) , . . . , ( G r , A r , B r ) of Hamil to nian s − t P a th in Bip ar tite Graphs such that | A i | = n A , | B i | = n B with n B = n A + 1 for i ∈ [ r ] outputs in p olynomial time an instance ( G ∗ , X ∗ ) of Hamil tonian Cycle p arameter ized by a modula tor to Ou terplanar graphs whic h is ye s if and only if one of the in put instances is yes , and such th at | X ∗ | is p olynomial in the size of the largest input instance plu s log r . The constru ct ion is s imila r to that of th e graph D ∗ in Theorem 7, with the diﬀerence that we are now u s ing un directed graphs and th at we use an outerplanar gadget to simulate directions of arcs. Assu me A i = { a i, 1 , . . . , a i,n A } and B i = { b i, 1 , . . . , b i,n B } for i ∈ [ r ]. W e b uild a graph G ∗ as follo ws. 20 1. F or i ∈ [ r ], for j ∈ [ n A ] add a cop y O i,j of the domino graph (Fig. 2) to G ∗ and lab el its terminals by a − i,j , a + i,j , ˆ a − i,j and ˆ a + i,j . 2. As the n ext step w e add edges to connect adjacen t d ominos. F or i ∈ [ r ], for j ∈ [ n A − 1] add the edge { a + i,j , a − i,j +1 } . 3. F or eac h ins ta n ce i ∈ [ r ] add three sp ecial v ertices x i , y i , and w i , together with edges { w i , x i } , { x i , a − i, 1 } , { a + i,n A , y i } . F or i ∈ [ r − 1] add the edges { y i , w i +1 } . 4. Ob serv e that at this stage, G ∗ ∈ Outerplanar . All ve r tic es w e add fr om this p oint on w ill go in to the mo dulator X ∗ suc h th at G ∗ − X ∗ will b e a mem b er of Outer planar . 5. W e add three sp ecial v ertices z − , z , z + with edges with { z − , z } an d { z , z + } . F u rthermore, w e add edges { x i , z − } and { z + , y i } for i ∈ [ r ]. 6. F or j ∈ [ n B ] add a v ertex b ∗ j to the graph G ∗ , and let B ∗ b e the set of these v ertices. Add edges { y r , b ∗ 1 } and { b ∗ n B , w 1 } . 7. As the last step of the c ons tr uctio n w e re-encod e th e b eha vior of the input graphs D i in to the instance. F or i ∈ [ r ], for eac h edge { a i,j , b i,f } in E ( G i ) add the edges { ˆ a − i,j , b ∗ f } and { ˆ a + i,j , b ∗ f } to G ∗ . Th is conclud es the description of G ∗ . No w deﬁn e X ∗ := { z − , z , z + } ∪ B ∗ . The outpu t of the cross-comp ositi on is the instance ( G ∗ , X ∗ ) of Hamil tonian Cycl e p arameterized by a modula tor to Ou terplanar graphs . It is easy to v erify that G ∗ − X ∗ ∈ Outer planar , and that th e constru ction can b e carried out in p olynomial time. The parameter | X ∗ | is b ounded by 1 + n B whic h is suﬃcien tly small. It remains to pro v e that G ∗ is yes if and only if one of the input instances is yes . W e ﬁrst p ro v e some rele v ant prop erties of G ∗ . Claim 2. L et C ⊆ E ( G ∗ ) b e a Hamiltonian cycle in G ∗ . The fol lowing must hold. 1. If C uses an ˆ a - tr aversal of some domino O i,j for i ∈ [ r ] and j ∈ [ n A ] then ther e ar e distinct indic es f , f ′ such that b f , the vertic es of the domino, and b f ′ app e ar c onse cutively on C . 2. If { x i , a − i, 1 } is not an e dge on C for some i ∈ [ r ] , then none of the e dges { a + i,j , a − i,j +1 } for j ∈ [ n A − 1] ar e c ontaine d in C , nor is the e dge { a + i,n A , y i } . 3. Ther e is an index i ∗ ∈ [ r ] suc h that { x i , a + i, 1 } is not use d on C . Pr o of. W e prov e the diﬀerent comp on ents consecutiv ely . 1. S upp ose C us es an ˆ a -tra v ersal of some d omino O i,j . By construction of G ∗ w e kno w that the only neigh b ors of ˆ a + i,j and ˆ a − i,j whic h are not con tained in the domino are of the form b ∗ f for some f ∈ [ n B ]. Since all ve r tic es of the d omin o are used on the cycle C in an ˆ a -tra v ersal (see Fig. 2) and the vertices ˆ a + i,j and ˆ a − i,j eac h h av e only one n eig hb or on C within the domino, eac h of th ese vertices m ust h av e a neigh b or outsid e the domino on C and hen ce these must b e of the form b ∗ f and b ∗ f ′ . W e must ha ve f 6 = f ′ otherwise w e w ould hav e a closed cycle con taining the domino O i,j and a single vertex b ∗ f , and this cycle would n ot b e Hamiltonian since it would not visit the ve rtex z . 21 2. Assu me th at { x i , a − i, 1 } is n ot on C , but at least one of the edges in the set Z i := {{ a + i,j , a − i,j +1 ) | j ∈ [ n A − 1] } ∪ {{ a + i,n A , y i }} is used on C . Let j ∗ b e sm allest in dex such that the vertex a + i,j ∗ is the end p oint of an edge in C ∩ Z i . Since a + i,j ∗ is endp oin t of an edge in C and the other endp oin t of th is edge do es not lie in the domin o O i,j ∗ , it follo ws that C con tains at most one edge in the d omin o O i,j ∗ whic h is inciden t on a + i,j ∗ . This ru les out the p ossibilit y that C mak es an ˆ a -tra versal of O i,j ∗ , since that r equires t wo edges within the d omino in ciden t on a + i,j ∗ . Because the construction of G ∗ together with P r oposition 5 ensu r es there are only t wo p ossible tra v ersals of the domino, we know that C must us e an a -tra ve r sal of O i,j ∗ . Th is tra versal uses exac tly one edge of the d omino incident on a − i,j ∗ . S ince a Hamiltonia n cycle m ust con tain exactly tw o edges incident on ev ery v ertex, this sho ws th at C must con tain some edge in ciden t on a − i,j ∗ whic h is n ot in the domino O i,j ∗ . But by construction of G ∗ there is only one suc h edge: if j ∗ = 1 then this edge is { x i , a − i, 1 } and otherw ise this edge is { a + i,j ∗ − 1 , a − i,j ∗ } . But b y the assump tio n at the s ta r t of the p roof and our c hoice of j ∗ , this edge is not con tained in C . Hence there is only o n e edge incident on a − i,j ∗ con tained in C , con tradicting the Hamiltonicit y of C . 3. Th e Hamiltonian cycle C must con tain t wo edges inciden t on ev ery ve r te x. By construction of G ∗ the v ertex z − is incident on the edge { z − , z } and on edges { x i , z − } f or i ∈ [ r ]. Hence if C con tains tw o edges incident on z − then at least one is of the form { x i ∗ , z − } for i ∗ ∈ [ r ]. Now consider the vertex w i ∗ , whic h has d eg r ee t wo by construction. T herefore b oth its incident edges m ust b e con tained in C , and in particular { w i ∗ , x i ∗ } is con tained in C . S o C cont ains t wo edges incident on x i ∗ whic h are unequal to the edge { x i ∗ , a − i ∗ , 1 } and since only tw o edges inciden t to eac h v ertex are con tained in C , it follo ws that C do es not conta in { x i ∗ , a − i ∗ , 1 } . Armed with these claims we p ro v e that G ∗ has a Hamiltonian cycle if and only if one of the input instances G i ∗ has a Hamiltonian path s ta r tin g in b 1 and ending in b n B . Firs t assume G ∗ has a Hamiltonian cycle C ⊆ E ( G ∗ ). By (3 ) th ere is an index i ∗ ∈ [ r ] suc h that { x i , a − i ∗ , 1 } 6∈ C . By (2) this implies there are no edges { a + i ∗ ,j , a − i ∗ ,j +1 } for j ∈ [ n A − 1] cont ained in C , nor is the edge { a + i ∗ ,n A , y i } . This sho ws that for eac h v ertex a − i ∗ ,j the only edges incident on a − i ∗ ,j whic h can b e con tained in C , are those of the d omino O i ∗ ,j whic h implies b y Prop osition 5 that C must do an ˆ a -tra v ersal of all dominos O i ∗ ,j for j ∈ [ n A ]. By (1 ) eac h tra ve r s al of a domino is preceded and succeeded by distinct v ertices b ∗ f , b ∗ f ′ and by constr u ctio n of G ∗ this can only o ccur if { a i ∗ ,j , b i ∗ ,f } and { a i ∗ ,j , b i ∗ ,f ′ } are edges of G i ∗ . Since v ertices b i ∗ , 1 and b i ∗ ,n B ha ve d eg r ee one in G i ∗ b y deﬁn iti on of Hamil tonian s − t P a th in Bip a r tite Graphs , and since n B = n A + 1, it now follo ws by reasoning similar to that of Theorem 7 that the set C i ∗ := {{ a i ∗ ,j , b i ∗ ,f } | b ∗ f precedes or su cceeds domino O i ∗ ,j on C } is a Hamiltonian p at h in G i ∗ b et w een v ertices b i ∗ , 1 and b i ∗ ,n B . F or the reve rs e direction, assume that G i ∗ has a Hamiltonian path C i ∗ starting in b i ∗ , 1 and ending in b i ∗ ,n B . Con s truct a Hamiltonian cycle in G ∗ as follo ws. • F or i 6 = i ∗ add the edges { w i , x i } , { x i , a − i, 1 } , { a + i,n A , y i } , and { a + i,j , a − i,j +1 } for j ∈ [ n A − 1] to C . • F or i 6 = i ∗ add the edges on the a -tra v ersals of all domin os O i,j with j ∈ [ n A ] to C . • Add the edges { w i ∗ , x i ∗ } , { x i ∗ , z − } , { z − , z } , { z , z + } , and { z + , y i ∗ } to C . Also add all edges on ˆ a -tra v ersals of the dominos O i ∗ ,j for j ∈ [ n A ] to C . • Add all ed ge s { y i , x i +1 } for i ∈ [ r − 1] to C . 22 • Add th e edges { b ∗ n B , w 1 } and { y r , b ∗ 1 } to C . • F or j ∈ [ n A ], if the edges of G i ∗ inciden t on a i ∗ ,j in the Hamiltonian p ath C i ∗ are { a i ∗ ,j , b i ∗ ,f } and { a i ∗ ,j , b i ∗ ,f ′ } then add the edges { ˆ a − i ∗ ,j , b ∗ f } and { ˆ a + i ∗ ,j , b ∗ f ′ } to C . (The ordering of b ∗ f and b ∗ f ′ is immaterial since th e domino can b e tra v ersed in either direction.) Using the constru ct ion of G ∗ is it straight -forward to verify that C is a Hamiltonian cycle in G ∗ , whic h p ro v es the rev erse d irecti on of the equiv alence and concludes th e pro of. 6 Finding paths with resp ect to forbidden pairs In this section w e study m ultiple parameterizations of sev eral v arian ts of path prob lems inv olving forbidden pairs. The ﬁrst version we consider is deﬁ n ed as f oll ows. s − t P a th with F orb i d den P airs P aramet erized by a Ver tex Cover of G Input: A graph G , d istinct vertic es s , t ∈ V ( G ), a set H ⊆  V ( G ) 2  of forbidden pairs, and a verte x co v er X of G . P arameter: ℓ := | X | . Question: Is there an s − t path in G w hic h con tains at m ost one v ertex of eac h pair { u, v } ∈ H ? W e can giv e evidence that this problem is not ﬁxed-parameter tractable. Theorem 9. s − t P a th with F orb idden P airs P aramete rized by a Ver t ex Cover o f G is har d for W[1]. Pr o of. W e give a parameterized r eduction f rom th e W[1]-hard k -Mul ticolore d Clique prob- lem [19]. Let ( G, k , φ ) b e an in stance of k -Mul ticolored Clique , wh ere φ : V ( G ) → { 1 , . . . , k } assigns eac h v ertex of G a color from 1 to k and the task is to d ecide whether G con tains a clique with one verte x of eac h of the k colors. W e construct a graph G ′ as follo ws. W e ﬁ rst mak e an indep endent set on the v ertices V ( G ), and add a forbidden p air for eac h p air of v ertices fr om V ( G ) that are either not adjacen t in G or w hic h h a v e the same color according to φ . T hen w e add k + 1 v ertices v 0 , v 1 , . . . , v k whic h will form the verte x co v er. W e connect v ertex v 0 to all ve r tic es of th e indep endent s et that ha ve color 1 according to φ . F or all i ∈ { 1 , . . . , k − 1 } , we connect v i to all v ertices that hav e colors i or i + 1. Finally , w e connect v k to all ve r tices of color k . W e set s := v 0 and t := v k , and r eturn the instance ( G ′ , v 0 , v k , X ′ := { v 0 , . . . , v k } ). W e n ot e that this can b e easily p erformed in p olynomial time, and that the parameter v alue, i.e., k + 1, is b ound ed b y a function of k . It is easy to see that ev ery s − t path has 2 k + 1 v ertices, and u ses all vertic es v 0 , . . . , v k as w ell as one ve r tex of eac h color. Note th at the colors are n ot part of the pr odu ce d in s ta n ce , b ut the forb idden pairs are used to enco de th is prop ert y . The latter vertices can also not b e non- adjacen t in G , and h ence they corresp ond to a m ulticolored clique of size k . S imilarly , giv en su ch a multicolo red clique it is straigh tforwa rd to ﬁnd an s − t path of length 2 k + 1 in G ′ that r espects the forbid d en pairs. Th u s W[1]-hardness of s − t P a th w i t h Forbidden P airs P arameter i z ed by a Ver tex Co ve r of G follo ws. 23 Let u s no w consider s ome v ariat ions of the pr oblem. Th e p roblems Shor test s − t P a th W ith F orbidd en P airs and Lon gest s − t P a t h With Forbidden P a irs are deﬁned similarly as s − t P a th with Forbidden P airs , with th e diﬀeren ce that there is an extra int eger k in the input and w e are asking for an s − t p at h con taining at most or at least k ve rtices. In Longest P a th With Forbidden P airs w e omit the inp uts s and t , and are lo oking for any su ﬃcien tly long path, regardless of its endp oint s. The related pr oblem Shor test P a th W ith Forbidden P airs is not in teresting, since its solution alw a ys consist of a p ath conta in in g a single vertex. It can b e easily v eriﬁed that asking for a p at h on at least 2 k + 1 v ertices, regardless of its endp oin ts, in the construction for the pro of of Th eo r em 9 leads also to a path whose v ertices from the indep enden t set made of V ( G ) corresp ond to a m ulticolored clique in G . Also, since asking f or a long or sh ort s − t path is as least as hard as asking for the existence of an s − t path, w e obtain the follo wing results as a corollary . Corollary 5. The pr oblems Sho r test s − t P a th W ith F orbidden P airs , Longest s − t P a th With F orbidd en P airs and Longest P a th With F orbidde n P airs , ar e har d for W[1] when p ar ameterize d by a vertex c over of G . Clearly , the hardness of the path p roblems with f orbidden pairs stems from the extra structure of th e forbid den pairs H , whic h is not taken int o account when consid ering structural parameters of G . In the follo wing w e consider the eﬀect of p aramete rizing by the structure of the graph G ∪ H (i.e., G with an add ed edge for ev ery forb idden pair). Using the optimization v ersion of Courcelle’s Theorem applied to structur es of b ounded tree- width [2 , 4, 8, 12, 13 ] (instead of graphs, as is more common) it is not d iﬃ cu lt to obtain an FPT result parameterized by the treewidth of G ∪ H . If we tak e a stru ct u re on the base set V ( G ) ∪ E ( G ) ∪ H whic h enco des an instance through un ary pr edica tes wh ic h sa y w hether an ob ject is a v ertex of G , edge of G , or forb idden pair in H , and through bin ary predicates which giv e the incidence b et w een v ertices and edges or pairs, then th e treewidth of the stru ct u re equals th e treewidth of G ∪ H . F or suc h a representa tion it is well-kno wn how to devise an MSOL formula which asks whether there exists a set of edges whic h forms a path b et w een s and t , and suc h that no tw o vertice s in ciden t on suc h an edge form a forbid den pair. Using standard extensions of MS O L w e ma y also maximize or minimize the size of a set of edges w hic h forms an s − t path resp ecting forbidden pairs, obtaining the follo wing result. Prop osition 6. The pr oblems Shor test s − t P a th W ith Forbidden P a irs , Longest s − t P a th With F orbidden P airs , and Longes t P a th With Forbidden P airs , ar e ﬁxe d- p ar ameter tr actable p ar ameterize d by the tr e ewidth of G ∪ H . F or the case of S hor test s − t P a th W i t h F orbidd en P airs the structure of G is actually not so imp ortant for the complexity of the problem: it is suﬃcient to p aramet erize by a verte x co v er of the graph on th e edge set H to obtain ﬁxed-p aramet er tractabilit y , as demons trat ed by the follo wing theorem. Theorem 10. Shor test s − t P a t h W ith Forbidden P airs P ar ameterized by a Ver tex Co ve r of H is ﬁxe d-p ar ameter tr actable. Pr o of. Giv en a graph G with endp oin ts s, t and forbidden pairs H s u c h that X is a vertex co ver of H , w e describ e ho w to ﬁn d the shortest s − t path whic h a v oids forbid den p ai r s. F or all su bsets X ′ ⊆ X whic h do not con tain a forb id den pair, we compute the shortest s − t path wh ic h inte r s ec ts X only 24 in X ′ as follo ws. Let Y b e the v ertices which form a f orbidden p air with a memb er of X ′ : then the d esired path is a sh ortest s − t path in the graph G − ( X \ X ′ ) − Y . Since a sh ortest path in this graph can b e foun d in linear time using breadth-ﬁrs t searc h, w e solv e the problem in O ∗ (2 | X | ) time by retur ning the shortest s − t path found o ver all c hoices of X ′ ⊆ X . It is easy to see that the p ositiv e news of Th eo r em 10, mem b ersh ip in FPT parameterized b y a vertex co ver of H , cannot b e extended to Longest s − t P a th With Forbidden P airs sin ce the latter pr oblem is already NP-complete when there are n o forbidden pairs. W e men tion without pro of that s − t P a th with Forbidden P airs is NP-complete when the graph ind uced by H is a matc hing, sho w in g that w e cannot improv e the parameterization by a vertex co v er of H to the treewidth of H . As the ﬁnal topic of this section we will consider the k ern eliz ation complexit y of forbidden path problems, obtaining a sup er-p olynomial lo wer b oun d on the k ernel size when p aramet erizing by a v ertex co v er of G ∪ H . Theorem 11. s − t P a th with Forbidden P airs P aramete rized by a Ver tex Cover of G ∪ H admits no p olynomial kernel unless NP ⊆ c oNP / p oly. Pr o of. W e consider the classical p roblem s − t P a t h with F orb idden P airs where we simp ly drop the set X from the p roblem description. W e show that s − t P a th with Forbidden P a irs cross-comp oses into s − t P a th with Forbidden P airs P arameterize d by a Ver tex Co ver of G ∪ H , wh ich suﬃces to p ro v e the claim by Theorem 1 since the construction of Th eo r em 9 sho ws th at s − t P a t h with Forbidden P airs is NP-complete. Let R b e a p olynomial equiv alence relation under which t wo well-formed ins tances ( G 1 , s 1 , t 1 , H 1 ) and ( G 2 , s 2 , t 2 , H 2 ) are equ iv alen t if | V ( G 1 ) | = | V ( G 2 ) | . W e sho w ho w to comp ose a sequence of inputs ( G 1 , s 1 , t 1 , H 1 ) , . . . , ( G r , s r , t r , H r ) on n vertice s eac h. Assume th e ve r tic es of V ( G i ) are la- b eled v 1 , . . . , v n suc h that v 1 = s i and v n = t i . W e build a graph G ∗ with a verte x cov er X ∗ , an d a set of forbidd en pairs H ∗ ⊆  V ( G ∗ ) 2  suc h th at all forbidden pairs hav e at least one mem b er in X ∗ . 1. F or j ∈ [ n ] add a v ertex v ∗ j to G ∗ . 2. Add a sp ecial starting v ertex w to G ∗ . 3. F or i ∈ [ r ] add a v ertex z i to G ∗ and mak e it adjacen t to w and v ∗ 1 . 4. F or 1 ≤ j < h ≤ n , add a ladder gadget L j,h with n sp ok es to G ∗ : • Ad d v ertices t i j,h and f i j,h to G ∗ for i ∈ [ n ]. • Ad d the set of ed ge s {{ t i j,h , t i +1 j,h } , { t i j,h , f i +1 j,h } , { f i j,h , f i +1 j,h } , { f i j,h , t i +1 j,h } | i ∈ [ n − 1] } to form the ins id e of the ladder. • Make v ∗ j adjacen t to t 1 j,h and f 1 j,h , and mak e v ∗ h adjacen t to t n j,h and f n j,h . 5. Rep eat the follo wing for eac h i ∈ [ r ]: • F or 1 ≤ j < h ≤ n , if { v j , v h } 6∈ E ( G i ) then add {{ z i , x } | x ∈ L j,h } to the set of forbidden pairs. Here L j,h denotes the set of 2 n v ertices on the ladder for { j, h } . • F or 1 ≤ j < h ≤ n , if { v j , v h } ∈ E ( G i ) then do as follo ws for all q ∈ [ n ]: – If { v j , v q } ∈ H i or { v h , v q } ∈ H i then add { z i , f q j,h } to H ∗ . 25 w z 1 z 2 z 3 v ∗ 1 v ∗ 2 v ∗ 3 v ∗ 4 t 1 1 , 2 f 1 1 , 2 t 2 1 , 2 f 2 1 , 2 t 3 1 , 2 f 3 1 , 2 t 4 1 , 2 f 4 1 , 2 Figure 3: An example of the lo wer-b ound construction of Th eorem 11, cross-comp osing three instances w ith n = 4 in to one. F or clarit y , only the vertice s of the ladder L 1 , 2 are d ra wn; the other ladders are visu al ized b y dotted paths. The forbid den pairs are dra wn as zigzagge d lines. F orb idden pairs with one elemen t in { z 1 , z 2 } are not dra wn. F or this example, the third input ( G 3 , s 3 , t 3 , H 3 ) has a forbidden pair { v 1 , v 3 } ∈ H 3 causing the forbidden pair { z 3 , f 3 1 , 2 } ∈ H ∗ . The v ertices b elo w the horizont al dashed line form th e vertex co v er X*. – Else, if { v j , v q } / ∈ H i and { v h , v q } / ∈ H i , then add { z i , t q j,h } . • F or 1 ≤ j < h ≤ n , for q ∈ [ n ], add { t q j,h , v ∗ q } to H ∗ . 6. Th is concludes the description of G ∗ and H ∗ , which is illustrated in Fig. 3. Deﬁne X ∗ := V ( G ∗ ) \ { z i | i ∈ [ r ] } . It is easy to see that th is construction can b e carried out in p olynomial time. The set X ∗ is a v ertex co v er for G ∗ ∪ H ∗ since all edges and forbidd en pairs of G ∗ ha ve at least one element in X ∗ . The size of X ∗ is 1 + n + 2 n  n 2  , whic h is p olynomial in n and therefore the p aramet er ℓ := | X ∗ | of the constructed instance is suitably small. W e set s ∗ := w and t ∗ := v ∗ n . Let us establish some prop erties of the constructed in stance . Claim 3. Consider an s ∗ − t ∗ p ath P in G ∗ which r esp e cts the forbidden p airs H ∗ , and orient the p ath such that it starts at s ∗ . The fol lowing must hold. 1. Ther e is exactly one index i ∗ ∈ [ r ] suc h that z i ∗ ∈ P , and the p ath P has the form ( s ∗ = w, z i ∗ , v ∗ 1 , . . . , v ∗ n = t ∗ ) . 2. If P c ontains v ∗ j and v ∗ h then { v j , v h } 6∈ H i ∗ , wher e i ∗ is as deﬁne d ab ove. 3. If v ∗ j and v ∗ h ar e vertic es on P , and no vertic es of the form v ∗ f ar e visite d by P b etwe en visiting v ∗ j and v ∗ h , then { v j , v h } ∈ E ( G i ∗ ) , wher e i ∗ is as deﬁne d ab ove. Pr o of. (1) Since all vertices z i for i ∈ [ r ] hav e the same neighborh oo d consisting of w and v ∗ 1 , if a path con tains at least tw o such v ertices then at least one of them is an end p oint of the path, wh ic h is n ot p ossible sin ce P is an s ∗ − t ∗ path. An y s ∗ − t ∗ path m ust use at least one vertex z i ∗ since all neigh b ors of s ∗ = w h a v e this f orm. (2) Supp ose P con tains v ∗ j and v ∗ h , an d assume for a cont r ad iction that { v j , v h } ∈ H i ∗ . Assume without loss of generalit y that j < h and th erefore v ∗ j 6 = v ∗ n . Since v ∗ j is not an endp oint of P , 26 v ertex v ∗ j m ust ha ve t wo neighb ors on P . By construction of G ∗ it easily follo ws th at at least one of th ese neighbors m ust lie on a ladder. Assume that v ∗ j has a neigh b or on a ladder L j,p ; th e case that the neigh b or lies on a ladder L p,j is symmetric. The forb idden pairs in volving z i ∗ and th e v ertices of the ladder L j,p ensure that for eac h sp ok e of the ladder on vertic es { t q j,p , f q j,p } th ere is a f orb idden pair con taining z i ∗ and exactly one vertex of the p ai r . Sin ce P conta ins z i ∗ it must a v oid exactly one v ertex of eac h sp ok e of the ladder, which implies b y construction of G ∗ that all v ertices on the ladd er ha ve only one v alid option along whic h to conti nue the p at h , implying that P m ust tra v erse the en tire ladder to the other end p oi nt v ∗ p . Sin ce { v j , v h } ∈ H i ∗ the deﬁn iti on of G ∗ sho ws th at { z i ∗ , f h j,p } ∈ H ∗ , and therefore path P must visit the other vertex t h j,p of the sp ok e wh en tra v ersin g the ladd er. But by the last step of the construction we kno w that { t h j,p , v ∗ h } ∈ H ∗ is a forbidden pair of wh ich P cont ains b oth v ertices; a con tradiction. (3) Supp ose that P successiv ely visits the v ∗ -v ertices v ∗ j and v ∗ h with j < h . By constru ction of G ∗ it is easy to see that P m ust trav erse the v ertices of the ladd er L j,h . Sin ce P cont ains z i ∗ , and z i ∗ forms a forbidden pair with ev ery v ertex on the ladder L j,h if { v j , v h } 6∈ E ( G i ∗ ), it follo ws that if P can trav erse the ladder L j,h without hitting a forbidden pair then the corresp ondin g edge m ust b e cont ained in G i ∗ . With these structural pr operties of the constructed instance we can pro ve the correctness of the cross-comp ositi on. Assum e that G ∗ is a yes -instance, and let P b e an s ∗ − t ∗ path in G ∗ . By (1) path P has the form ( w , z i ∗ , v ∗ 1 , . . . , v ∗ n ). Orient P from s ∗ to t ∗ , and consider the ver- tices ( v ∗ i 1 , . . . , v ∗ i m ) in the s et { v ∗ i | i ∈ [ n ] } which are successiv ely visited on this p ath. By (2) it follo ws that there are n o ind ices i j , i f suc h that { v i j , v i f } ∈ H i ∗ , and b y (3 ) th e set E ( G i ∗ ) con tains all edges { v i j , v i j +1 } f or j ∈ [ m − 1]. Hence the vertice s ( v i 1 , . . . , v i m ) form a path in G i ∗ whic h do es not contai n a forbidden p air. By the form of P describ ed in (1) it follo ws that v ∗ i 1 = v ∗ 1 , and since P ends with v ∗ n it follo ws that v ∗ i m = v ∗ n . Hence this suggested path is a v 1 − v n path in G i ∗ whic h av oids forb idden p airs, w hic h pro ves that instance num b er i ∗ is yes . F or the r ev erse dir ection, supp ose that G i ∗ con tains a path ( v i 1 , . . . , v i m ) with i 1 = 1 and i m = n whic h do es not con tain a forb id den pair. No w construct a path in G ∗ whic h starts at s ∗ = w and successiv ely visits z i ∗ and v ∗ 1 . It then tra v erses the ladder to v ∗ i 2 , and b y construction of G ∗ there is exactly one vertex on eac h sp ok e of the ladd er which is not in a forbidden p air with z i ∗ ; the path uses these v ertices. W e can con tinue this, alternatingly tra versing a v ertex v ∗ i j and a ladder, unt il reac hing v ∗ i m = t ∗ . It is not diﬃcult to verify that the constructed p at h do es not con tain an y forbidden pairs of H ∗ , whic h concludes the pr oof. It is easy to mo dify the construction of Theorem 11 to pr o v e a k ernel lo we r b ound for Lon gest P a th With F orbidden P airs P arame terized by a Ver tex Cover of G ∪ H . The deﬁnition of the latte r problem does not allo w us to sp ecify th e endp oints s and t of the p at h , but b y creating t wo suitably long p aths and connecting th ese only to s and t w e can ensure that a suﬃcien tly long path in the resulting graph is actually an s − t path. Also, similarly as b efore the hard ness results for the existence problem of s − t paths im m ediat ely carry o ver to long and short s − t paths. W e obtain the follo wing results as a corollary . Corollary 6. The pr oblems Shor test s − t P a th With Forbidden P airs , Longes t s − t P a th With F orbidde n P airs , and Longes t P a t h With Forbidden P airs , do not admit a p olynomial kernel p ar ameterize d by a vertex c over of G ∪ H unless NP ⊆ c oNP / p oly. T able 1 con tains a summary of the results of this section. 27 vc ( G ) vc ( H ) tw ( H ) tw ( G ∪ H ) vc ( G ∪ H ) s − t P a th F.P. W[1]-hard FPT P ara-NP-c FPT No p oly Shor test s − t P a th F.P. W[1]-hard FPT P ara-NP-c FPT No p oly Longest s − t P a th F.P. W[1]-hard P ara-NP-c P ara-NP-c FPT No p oly Longest P a th F.P. W[1]-hard P ara-NP-c P ara-NP-c FPT No p oly T able 1: Comp le xity o verview of p ath problems with forbidd en pairs. Eac h column r ep resen ts a diﬀeren t p arameterization; vc ( G ) denotes the minim u m ve rtex co v er size of G , and tw ( G ) denotes its treewidth. F.P . abbreviates “ with F or bidden P airs ”. Th e classiﬁcation “N o p oly” means “no p olynomial k ernel unless NP ⊆ coNP / p oly” , and “P ara-NP-c” means “NP-complete for a constant v alue of the p aramet er”. F or a parameterization in FPT, w e either list “FPT” or “No p oly”, dep ending on w hic h of the t wo is more relev ant: all problems listed as “No p oly” are in FPT, and no problem listed as “FPT” admits a p olynomial ke rn el. Shor test P a th F.P. is trivially in P . 7 Conclusion In th is wo r k w e hav e sho wn that for su ﬃcien tly strong stru ctural parameterizations, man y p at h and cycle p r oblems admit p olynomial kernels ev en though their natural parameterizations do not. The marking te chnique using bipartite matc hing yields quadratic-v ertex k ernels f or many problems parameterized by the size of a v ertex cov er. W e in tro duced a binary enco ding tric k w hic h giv es p olynomial kernels for p roblems parameterized by the max leaf num b er or by a mo dulator to cographs. On th e n eg ative side, w e also exhibited smaller s tr uctural parameters whic h pro v ably do not lead to p olynomial kernels for Hamil tonian Cycle u nless NP ⊆ coNP / p oly. Let us reﬂect brieﬂy on the parameters used for the upp er- and lo wer b ounds. Recall that the v ertex co ver n u m b er of a graph can also b e in terpreted as the num b er of ve r tex- deletions needed to reduce the graph to an indep endent set, i.e. the vertex-dele tion d istance to a graph of treewidth 0. Hence Theorem 3 sho w s that Long Cycle admits a p olynomial kernel parameterized by ve r te x-d eletion distance to treewidth 0. On the other hand , Theorem 8 sh o ws that if NP 6⊆ coNP / p oly th en Hamil tonian Cycle do es not h a v e a p olynomial kernel parameterized b y the deletion d istance to treewidth tw o (since outerplanar graphs ha ve treewidth at most t wo ), and of course th is carries o v er to the h arder problem Long Cycle . It is in teresting to settle wh at happ ens for treewidth one, i.e., forests: do es Hamil tonian Cycle parameterized by a feedb ac k v ertex set admit a polynomial k ernel? T o generalize the r esu lt of Theorem 6 by distance to a cluster graph, one could consider th e distance to cographs. 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Kernel Bounds for Path and Cycle Problems

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