Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs

Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs Noga Alon 1 and Shai Gutner 2 1 Schools of Mathematics and Computer Science, T el-Aviv Universit y , T el-Aviv, 69978, Israel. ⋆ noga@math. tau.ac.il. 2 School of Computer S cience, T el-Aviv Un iversi ty , T el-Aviv, 69978, I srael. ⋆⋆ gutner@tau .ac.il. Abstract. There is substantia l literature dealing with fix ed p arameter algorithms for the dominating set problem on v arious families of graph s. In this pap er, we give a k O ( dk ) n time algorithm for finding a dominat- ing set of size at most k in a d -degenerated graph with n ver tices. This prov es that the dominating set problem is fixed-parameter tractable for degenerated graphs. F or graphs that do not contain K h as a top ological minor, w e g ive an impro ved al gorithm for th e problem with running time ( O ( h )) hk n . F or graphs which are K h -minor-free, the run ning time is fur- ther reduced to ( O (log h )) hk/ 2 n . Fixed-parameter t ractable algorithms that are linear in the number of vertices of the graph were prev iously known only for planar graphs. F or t he families of graphs discussed ab ov e, th e problem of find in g an induced cycle of a giv en length is also ad d ressed. F or every fixed H and k , we show that if an H -minor-free graph G with n vertices contains an induced cy cle of size k , then such a cycle can b e found in O ( n ) exp ected time as well as in O ( n log n ) worst-case time. Some results are stated concerning the ( im)p ossibilit y of establishing linear time algorithms for the more general family of degenerated graphs. Key words: H-minor-free graphs, degenerated graphs, dominating set problem, finding an induced cycle, fixed -parameter tractable algorithms. 1 In tro duction This pap er deals with fixed-pa rameter algo r ithms for degene r ated gra phs. The degeneracy d ( G ) of an undirected g raph G = ( V , E ) is the sma llest num ber d for which ther e exists a n a cyclic or ie n tation of G in which all the outdegrees are at mos t d . Many interesting families of g raphs ar e degenera ted (hav e b ounded ⋆ Researc h supp orted in part by a grant from th e Israel Science F oundation, and by the Hermann Mink ow ski Minerv a Center for Geometry at T el Aviv Universit y . ⋆⋆ This pap er forms part of a Ph.D. t hesis written by the author under the sup ervision of Prof. N. Alon and Prof. Y . Azar in T el Aviv Universit y . degeneracy ). F or example, gr aphs embeddable on some fixed surface, degree- bo unded gr aphs, gra phs o f b ounded tr ee-width, and non-trivial minor- closed families of gr aphs. There is an extensive literature dealing with fixe d- parameter a lgorithms for the do mina ting set problem o n v arious families of graphs. O ur main result is a linear time algorithm for finding a do minating set of fixe d size in degener - ated gr a phs. This is the mos t general cla ss of graphs for which fixed- parameter tractability for this problem has b een establis hed. T o the b est of our knowledge, linear time algor ithms for the domina ting s et pr oblem were previously known only for planar gr aphs. Our alg orithms b oth gener alize and simplify the cla ssical bo unded sear ch tr e e a lgorithms for this problem (see, e.g., [2, 13]). The problem o f finding induced cycles in degenerated gr aphs has b een studied by Cai, Chan and Chan [8]. O ur seco nd result in this pap er is a rando mized algorithm for finding an induced cycle o f fixed size in gr aphs with an excluded minor. The alg orithm’s expec ted running time is linear, and its der a ndomization is done in an efficient wa y , answering a n op en questio n from [8]. The pro blem of finding induced cy c les in degener ated g raphs is also addre s sed. The Dominating Se t Proble m. The domina ting set pr oblem on gene r al graphs is known to b e W [2]-complete [12]. This means that most likely there is no f ( k ) · n c -algor ithm for finding a dominating set of size at most k in a gra ph of size n for any computable function f : I N → I N and constant c . This sugg e sts the explo ration of sp ecific families of graphs for which this pro ble m is fixed- parameter tra ctable. F or a general introductio n to the field of par ameterized complexity , the reader is referred to [12] a nd [14]. The metho d of b o unded search trees ha s been us e d to give an O (8 k n ) time algorithm for the do minating set problem in plana r gr aphs [2] and an O ((4 g + 40) k n 2 ) time algorithm for the problem in graphs of bounded genus g ≥ 1 [13]. The alg orithms for planar gra ph were improved to O (4 6 √ 34 k n ) [1], then to O (2 27 √ k n ) [17 ], a nd finally to O (2 15 . 13 √ k k + n 3 + k 4 ) [15]. Fixed-pa rameter algorithms ar e now known also for map graphs [9] and for co nstant pow ers of H - minor-free gr aphs [1 0 ]. The r unning time given in [1 0 ] for finding a dominating set of size k in an H -minor-free graph G with n vertices is 2 O ( √ k ) n c , where c is a constant dep ending o nly on H . T o s ummarize these res ults, fixed-para meter tractable algo rithms for the domina ting set proble m were known for fix e d p ow- ers of H -minor-free gra phs and for map g raphs. Line a r time algo rithms were established only for planar graphs. Finding Paths and Cycles. The founda tions for the algorithms for finding cycles, presented in this pa per , hav e be e n laid in [4], where the author s intro- duce the c o lor-co ding technique. Tw o main ra ndomized a lgorithms are present ed there, as follows. A simple directed or undirected path of length k − 1 in a gra ph G = ( V , E ) that contains such a pa th can b e found in 2 O ( k ) | E | ex pe c ted time in the dire c ted ca s e a nd in 2 O ( k ) | V | exp ected time in the undirected case. A simple directed or undirected cycle of size k in a graph G = ( V , E ) tha t c ontains such a cycle ca n b e found in either 2 O ( k ) | V || E | or 2 O ( k ) | V | ω exp ected time, wher e ω < 2 . 376 is the exp onent of ma trix m ultiplication. These algorithms can b e de- randomized at a cost of an extra lo g | V | factor. As for the case of ev en cycles, it is shown in [2 3 ] that for every fixed k ≥ 2 , there is an O ( | V | 2 ) algor ithm for finding a simple cycle o f s iz e 2 k in a n undir e cted gra ph (that co n tains such a cyc le). Improv ed algo r ithms for detecting given length cycles hav e b een presented in [5] and [24 ]. The a uthors of [5] describ e fast algor ithms for finding short cy c les in d -degenerated graphs. In pa rticular, C 3 ’s a nd C 4 ’s can b e found in O ( | E | · d ( G )) time and C 5 ’s in O ( | E | · d ( G ) 2 ) time. Finding Induced Pa ths and Cycles. Cai, Chan and Chan hav e recently int ro duced a new interesting technique they ca ll r andom sep ar ation for solv- ing fix ed-cardinality o ptimization problems on graphs [8]. They co mbin e this techn ique to gether with c o lor-co ding to give the following a lgorithms for find- ing an induced gra ph within a lar g e gr aph. F or fixed constants k and d , if a d -degenerated g raph G with n vertices contains some fixed induced tree T on k vertices, then it can b e found in O ( n ) exp ected time and O ( n log 2 n ) worst- case time. If such a gra ph G c o ntains an induced k -cycle, then it can b e fo und in O ( n 2 ) exp ected time a nd O ( n 2 log 2 n ) worst-case time. Two op en pro blems are raised by the a uthors of the pap er. First, they a s k whether the log 2 n fac- tor incurred in the der andomization can be re duce d to log n . A s econd question is whether there is an O ( n ) expe cted time a lgorithm for finding an induced k -cycle in a d -deg enerated gr aph with n vertices. In this pap er, we show that when combining the techniques o f random sepa ration and colo r-co ding, an im- prov ed derandomizatio n with a los s of only log n is indeed p ossible. An O ( n ) exp ected time algorithm finding an induced k -cy c le in graphs with an excluded minor is pres ent ed. W e give evidence that establishing such a n alg o rithm even for 2-degene r ated graphs has far-re a ching co ns equences. Our Results. The main res ult of the pap er is that the dominating set prob- lem is fixed-para meter tra ctable for deg enerated graphs. The running time is k O ( d k ) n for finding a dominating set of siz e k in a d -degener a ted gra ph with n vertices. The algo rithm is linear in the num b er of vertices o f the gra ph, and we further improve the dep endence on k for the following sp ecific families of degenerated graphs. F or gra phs that do not contain K h as a top olog ical mi- nor, an improv ed algor ithm for the problem with running time ( O ( h )) hk n is established. F or graphs which a re K h -minor-free, the running time obtained is ( O (log h )) hk/ 2 n . W e show that all the alg orithms can b e generalized to the weigh ted ca se in the following se nse. A domina ting set o f size at most k having minim um weigh t can b e found within the sa me time bo unds. W e a ddress tw o op en questions ra is ed by Cai, Chan a nd Chan in [8] concern- ing linear time alg o rithms fo r finding a n induced cy cle in degenerated gra phs. An O ( n ) ex p ected time algorithm for finding an induced k -cycle in graphs with an excluded minor is presented. The derandomizatio n p er formed in [8] is im- prov ed and we get a deterministic O ( n lo g n ) time algor ithm for the problem. As for finding induced cycles in degenera ted graphs, we show a deterministic O ( n ) time alg orithm for finding cyc le s of size a t most 5, a nd also explain why this is unlikely to b e po ssible to a chiev e for long er cycles. T ec h ni ques. W e gener alize the known sea rch tree alg orithms for the dom- inating set pro blem. This is enabled by pr oving s o me combinatorial lemmas, which ar e interesting in their own right. F or de g enerated gr aphs, we b o und the nu mber of vertices tha t dominate man y elements of a given set, wherea s for graphs with a n excluded minor , our interest is in vertices that still need to b e dominated and have a small degr e e. The a lgorithm for finding an induced cycle in no n- trivial mino r-closed fam- ilies is based o n random separa tion and co lo r-co ding. Its derandomiza tion is per formed using known e xplicit co nstructions o f families of (gener a lized) p er fect hash functions. 2 Preliminaries The pa per deals with undirected and simple graphs, unless stated o therwise. Generally speaking , we will follow the notation used in [7] and [11 ]. F or a n undirected gr aph G = ( V , E ) and a vertex v ∈ V , N ( v ) denotes the set o f all vertices adjacent to v (not including v itself ). W e sa y that v dominates the vertices of N ( v ) ∪ { v } . The graph obtained from G by deleting v is deno ted G − v . The subgr aph of G induced by some set V ′ ⊆ V is de no ted by G [ V ′ ]. A gra ph G is d -de gener ate d if every induced subg r aph of G has a vertex of degree at most d . It is easy and k nown that every d -degener ated g r aph G = ( V , E ) admits an a cyclic or ientation such that the outdeg ree o f each vertex is at most d . Such an orie ntation can be found in O ( | E | ) time. A d -degener ated gr aph with n vertices ha s less than dn edges a nd there fore its average degree is less than 2 d . F or a dir ected g raph D = ( V , A ) a nd a vertex v ∈ V , the set of out-neighbor s of v is deno ted by N + ( v ). F or a set V ′ ⊆ V , the notatio n N + ( V ′ ) stands for the set of all vertices that are out-neighbo rs of at least one vertex of V ′ . F or a directed gra ph D = ( V , A ) a nd a vertex v ∈ V , we define N + 1 ( v ) = N + ( v ) and N + i ( v ) = N + ( N + i − 1 ( v )) for i ≥ 2. An edge is said to b e sub divide d when it is deleted and replac e d by a path of length tw o connecting its ends, the internal vertex of this path b e ing a new vertex. A sub divisio n of a graph G is a g r aph that can b e obtained fro m G by a se q uence of e dg e sub divisions. If a sub division of a gra ph H is the subgr a ph of a nother graph G , then H is a top olo gic al minor of G . A g raph H is c alled a minor of a graph G if is can b e obta ined from a subgr aph o f G by a series of edge contractions. In the parameter iz ed dominating set problem, we are given an undire c ted graph G = ( V , E ), a pa r ameter k , and need to find a set of at mos t k v ertices that dominate all the other vertices. F ollowing the ter minology o f [2], the following generaliza tion o f the problem is considered. The input is a blac k and white gr aph , which simply means that the vertex set V o f the gra ph G has be en pa rtitioned int o tw o disjo int se ts B and W of black and white vertices, r esp ectively , i.e., V = B ⊎ W , wher e ⊎ denotes disjoint set union. Given a black and white g raph G = ( B ⊎ W, E ) and a n integer k , the problem is to find a set of at most k vertices that dominate the black v ertices. More forma lly , we a sk whether there is a subset U ⊆ B ⊎ W , such that | U | ≤ k and e very vertex v ∈ B − U s a tisfies N ( v ) ∩ U 6 = ∅ . Finally we give a new definition, s p ecific to this pap er , for what it means to be a r e duc e d bla ck and white gr a ph. Definition 1. A black and white gr aph G = ( B ⊎ W , E ) is c al le d r e duc e d if it satisfies the fol lowing c onditions: – W is an indep endent set. – A l l the vertic es of W have de gr e e at le ast 2 . – N ( w 1 ) 6 = N ( w 2 ) for every two distinct vertic es w 1 , w 2 ∈ W . 3 Algorithms for the Dominating Set Problem 3.1 Degenerated Graphs The alg o rithm for degenerated g r aphs is based o n the following combinatorial lemma. Lemma 1. L et G = ( B ⊎ W, E ) b e a d -de gener ate d black and white gr aph. If | B | > (4 d + 2) k , then ther e ar e at most (4 d + 2 ) k vertic es in G that dominate at le ast | B | /k vertic es of B . Pr o of. Denote R = { v ∈ B ∪ W   | ( N G ( v ) ∪ { v } ) ∩ B | ≥ | B | /k } . By co ntradiction, assume that | R | > (4 d + 2) k . The induced subgr aph G [ R ∪ B ] has at mo st | R | + | B | vertices and a t least | R | 2 · ( | B | k − 1) edges. The av erage degre e o f G [ R ∪ B ] is th us at least | R | ( | B | − k ) k ( | R | + | B | ) ≥ min {| R | , | B |} 2 k − 1 > 2 d. This contradicts the fact that G [ R ∪ B ] is d -degenerated. ⊓ ⊔ Theorem 1. Ther e is a k O ( d k ) n t ime algorithm for finding a dominating set of size at m ost k in a d -de gener ate d black and white gr aph with n vertic es that c ontains su ch a set. Pr o of. The pseudo co de o f algorithm D ominatin g S e tD eg e ner ated ( G, k ) that solves this pro blem a pp e a rs b elow. If ther e is indeed a dominating s et of size at mo st k , then this means that we can split B into k disjoin t piece s (some of them can b e empty), so that e a ch piece has a vertex that do minates it. If | B | ≤ (4 d + 2) k , then there are at most k (4 d +2) k wa y s to divide the set B int o k disjoint pieces . F or ea ch such split, w e can check in O ( k dn ) time whether every piece is dominated by a vertex. If | B | > (4 d + 2) k , then it follows fro m Lemma 1 that | R | ≤ (4 d + 2) k . This means that the search tree can grow to b e o f size at most (4 d + 2) k k ! b efor e po ssibly r eaching the previous ca se. This gives the needed time b ound. ⊓ ⊔ Algorithm 1 : D ominatin g S e tD eg e ner ated ( G, k ) Input : Black and white d -degenerated graph G = ( B ⊎ W, E ), integers k , d Output : A set dominating all vertices of B of size at most k or N O N E if no such set exists if B = ∅ the n return ∅ else if k = 0 then return N O N E else if | B | ≤ (4 d + 2) k then forall p ossible ways of splitting B into k (p ossibly empty) di sjoint pie c es B 1 , . . . , B k do if e ach pi e c e B i has a vertex v i that dominates it then return { v 1 , . . . , v k } return N O N E else R ← { v ∈ B ∪ W ˛ ˛ | ( N G ( v ) ∪ { v } ) ∩ B | ≥ | B | /k } forall v ∈ R do Create a new graph G ′ from G by marking all the elements of N G ( v ) as white and removing v from th e graph D ← Dom inating S etD eg ener ated ( G ′ , k − 1) if D 6 = N O N E then return D ∪ { v } return N O N E 3.2 Graphs wi th an Excluded Mi nor Graphs with either an excluded minor or with no top olo gical mino r are known to b e deg enerated. W e will apply the following useful prop os itions. Prop ositi o n 1. [6, 18] Ther e exist s a c onst ant c su ch that, for every h , every gr aph that do es not c ontain K h as a top olo gic al minor is ch 2 -de gener ate d. Prop ositi o n 2. [19, 21, 22] Ther e exist s a c onstant c such that, for every h , every gr aph with no K h minor is ch √ log h -de gener ate d. The following lemma gives an upp er b ound on the num b er of cliques of a prescrib ed fixed size in a degener ated g raph. Lemma 2. If a gr aph G with n vertic es is d -de gener ate d, t hen for every k ≥ 1 , G c ontains at most  d k − 1  n c opies of K k . Pr o of. By induction o n n . F or n = 1 this is obviously true. I n the g eneral case , let v b e a vertex of deg ree a t most d . The num b er of copies of K k that co nt ain v is at mos t  d k − 1  . By the induction hypothes is, the num b e r of copies of K k in G − v is at most  d k − 1  ( n − 1 ). ⊓ ⊔ W e ca n now prove o ur main combinatorial r esults. Theorem 2. Ther e exists a c onstant c > 0 , su ch that for every r e duc e d black and white gr aph G = ( B ⊎ W, E ) , if G do es not c ontain K h as a t op olo gic al minor, then ther e exists a vert ex b ∈ B of de gr e e at most ( ch ) h . Pr o of. Denote | B | = n > 0 and d = c h 2 where c is the co ns tant from Prop osition 1. Consider the vertices of W in s ome arbitrary o rder. F or each such vertex w ∈ W , if ther e exist tw o vertices b 1 , b 2 ∈ N ( w ), such tha t b 1 and b 2 are not connected, a dd the edge { b 1 , b 2 } and remo ve the v ertex w from the gra ph. Denote the res ulting gra ph G ′ = ( B ⊎ W ′ , E ′ ). Obviously , G ′ [ B ] do es no t contain K h as a top olog ical minor a nd therefore has at most dn edg es. The num b er of edges in the induced subgraph G ′ [ B ] is at least the num b er of white vertices that were deleted from the g raph, which means that at most dn w ere deleted s o far. W e now b ound | W ′ | , the num b er o f white vertices in G ′ . It follows fr o m the definition of a reduced black and white gra ph that there are no white vertices in G ′ of degree smaller than 2. The gra ph G ′ cannot contain a white vertex of degree h − 1 or more, since this would mean that the orig ina l gra ph G contained a sub division of K h . Now le t w be a white vertex o f G ′ of degree k , where 2 ≤ k ≤ h − 2. The reason why w was not deleted during the pro cess o f generating G ′ is bec ause N ( w ) is a clique of size k in G ′ [ B ]. The graph G ′ is a reduced black and white graph, and therefore N ( w 1 ) 6 = N ( w 2 ) for every tw o different white vertices w 1 and w 2 . This means that the neighbors of each white v ertex induce a differen t clique in G ′ [ B ]. B y a pplying Lemma 2 to G ′ [ B ], we g et that the nu mber of white vertices of degree k in G ′ is at most  d k − 1  n . This means that | W ′ | ≤ h  d 1  +  d 2  + · · · +  d h − 3  i n . W e know that | W | ≤ | W ′ | + dn and therefore | E | ≤ d ( | B | + | W | ) ≤ d h 3 d +  d 2  + · · · +  d h − 3  i n . Ob viously , there e x ists a black vertex o f degr e e at most 2 | E | /n . The result now follows b y plug ging the v a lue of d and using the fact that  n k  ≤ ( en k ) k . ⊓ ⊔ Theorem 3. Ther e exists a c onstant c > 0 , su ch that for every r e duc e d black and white gr aph G = ( B ⊎ W, E ) , if G is K h -minor-fr e e, then ther e exists a vertex b ∈ B of de gr e e at most ( c log h ) h/ 2 . Pr o of. W e pr o ceed as in the pro of o f Theo rem 2 using Pr op osition 2 instead of Prop ositio n 1. ⊓ ⊔ Theorem 4. Ther e is an ( O ( h )) hk n time algorithm for finding a dominating set of size at most k in a black and white gr aph with n vertic es and no K h as a top olo gic al minor. Pr o of. The pseudo co de of alg orithm D ominating S etN oM inor ( G, k ) that solves this pr o blem app ear s b elow. Let the input b e a bla ck and white gr aph G = ( B ⊎ W, E ). It is imp ortant to notice that the a lgorithm removes vertices and edges in or der to get a (nearly) reduced black and white graph. This can b e done in time O ( | E | ) by a car e ful pro cedure based on the proof of Theor em 2 co m bined with radix sorting. W e omit the details which will app ear in the full version o f the pap er. The time bo und for the a lgorithm now follows from Theorem 2. ⊓ ⊔ Algorithm 2 : D ominatin g S e tN oM inor ( G, k ) Input : Black and white ( K h -minor-free) graph G = ( B ⊎ W, E ), integer k Output : A set dominating all vertices of B of size at most k or N O N E if no such set exists if B = ∅ the n return ∅ else if k = 0 then return N O N E else Remov e all edges of G whose tw o endp oints are in W Remov e all white vertic es of G of degree 0 or 1 As long as th ere are tw o different vertices w 1 , w 2 ∈ W with N ( w 1 ) = N ( w 2 ) , | N ( w 1 ) | < h − 1, remo ve one of them from the graph Let b ∈ B b e a vertex of minimum degree among all ver tices in B forall v ∈ N G ( b ) ∪ { b } do Create a new graph G ′ from G by marking all the elements of N G ( v ) as white and removing v from th e graph D ← Dom inating S etN oM inor ( G ′ , k − 1) if D 6 = N O N E then return D ∪ { v } return N O N E Theorem 5. Ther e is an ( O (lo g h )) hk/ 2 n t ime algorithm for finding a domi- nating set of size at most k in a black and white gr aph with n vertic es which is K h -minor-fr e e. Pr o of. The pro of is analo gues to that of Theo rem 4 using Theore m 3 instead o f Theorem 2. ⊓ ⊔ 3.3 The W eighted Case In the weigh ted do minating set problem, each vertex of the gr aph has some po sitive real weight. The g oal is to find a dominating set of size a t most k , such that the sum of the weigh ts of all the vertices o f the dominating set is as small as p os s ible. The a lgorithms we presented can b e gener a lized to deal with the weigh ted case without changing the time b ounds. In this c a se, the who le search tree needs to b e sca nned and one cannot settle for the first v alid solution found. Let G = ( B ⊎ W , E ) b e the input gr aph to the algor ithm. In algo rithm 1 for degenerated gr aphs, we need to address the case where | B | ≤ (4 d + 2) k . In this case, the alg o rithm scans all p ossible wa ys o f s plitting B into k disjoint piec es B 1 , . . . , B k , and it ha s to b e mo dified, so that it will alwa ys cho o se a vertex with minimum weight that dominates each piece. In alg orithm 2 for graphs with an exc luded minor, the criter io n for removing white vertices from the gra ph is mo dified so that whenever tw o vertices w 1 , w 2 ∈ W satisfy N ( w 1 ) = N ( w 2 ), the vertex with the bigger weigh t is remov ed. 4 Finding Induced Cycles 4.1 Degenerated Graphs Recall that N + i ( v ) is the set of all vertices that ca n b e reached from v b y a directed path of length exac tly i . If the o utdegree of every vertex in a directed graph D = ( V , A ) is a t most d , then obviously | N + i ( v ) | ≤ d i for every v ∈ V and i ≥ 1. Theorem 6. F or every fix e d d ≥ 1 and k ≤ 5 , ther e is a deterministic O ( n ) time algorithm for finding an indu c e d cycle of length k in a d -de gener ate d gr aph on n vertic es. Pr o of. Giv en a d -deg enerated gr aph G = ( V , E ) with n vertices, we orie nt the edges so that the outdegree o f all vertices is at most d . This can be do ne in time O ( | E | ). Denote the resulting dir e cted graph D = ( V , A ). W e can further as sume that V = { 1 , 2 , . . . , n } and that every directed edge { u, v } ∈ A satisfies u < v . This means that an out-neighbor of a vertex u will a lwa ys have an index which is bigge r than that of u . W e now describ e how to find cyc le s of size at most 5. T o find cycles of size 3 we simply chec k for ea ch vertex v whether N + ( v ) ∩ N + 2 ( v ) 6 = ∅ . Suppo se now that we wan t to find a cy c le v 1 − v 2 − v 3 − v 4 − v 1 of size 4. Without loss of gener ality , a ssume that v 1 < v 2 < v 4 . W e distinguish betw een tw o po s sible cas e s. – v 1 < v 3 < v 2 < v 4 : Keep t wo co unters C 1 and C 2 for each pair of vertices. F or every vertex v ∈ V and every unordered pair o f distinct vertices u, w ∈ N + ( v ), such that u and w ar e not connected, w e raise the counter C 1 ( { u, w } ) by one. In addition to that, for every vertex x ∈ N + ( v ) such that u, w ∈ N + ( x ), the co unt er C 2 ( { u, w } ) is incremen ted. After completing this pro cess, we c heck whether there are tw o vertices for which  C 1 ( { u,w } ) 2  − C 2 ( { u, w } ) > 0. This would imply that an induced 4-cycle was found. – v 1 < v 2 < v 3 < v 4 or v 1 < v 2 < v 4 < v 3 : Check for each vertex v whether the set { v } ∪ N + ( v ) ∪ N + 2 ( v ) ∪ N + 3 ( v ) contains an induced cycle. T o find an induced cycle of size 5, a mo r e detailed case a na lysis is needed. It is easy to v erify that such a cycle has one of the following tw o t yp es. – There is a vertex v such tha t { v } ∪ N + ( v ) ∪ N + 2 ( v ) ∪ N + 3 ( v ) ∪ N + 4 ( v ) co ntains the induced cycle. – The cyc le is o f the for m v − x − u − y − w − v , where x ∈ N + ( v ), u ∈ N + ( x ) ∩ N + ( y ), and w ∈ N + ( v ) ∩ N + ( y ). The induced cycle can be found by defining c o unters in a similar w ay to what was done befor e. W e omit the details. ⊓ ⊔ The following s imple lemma shows that a linear time algorithm for finding an induced C 6 in a 2-degener a ted gra ph would imply that a triangle (a C 3 ) can be found in a ge ne r al graph in O ( | V | + | E | ) ≤ O ( | V | 2 ) time. It is a long standing op en question to impr ov e the natural O ( | V | ω ) time algo rithm for this pr oblem [16]. Lemma 3. Giv en a line ar t ime algorithm for finding an induc e d C 6 in a 2 - de gener ate d gr aph, it is p ossible to find triangles in gener al gr aphs in O ( | V | + | E | ) time. Pr o of. Giv en a g r aph G = ( V , E ), sub divide all the edg es. The new g raph ob- tained G ′ is 2-degener ated and has | V | + | E | vertices. A linear time algo rithm for finding an induced C 6 in G ′ actually finds a triangle in G . By a ssumption, the running time is O ( | V | + | E | ) ≤ O ( | V | 2 ). ⊓ ⊔ 4.2 Minor-Clos ed F amilies of Graphs Theorem 7. Supp ose that G is a gr aph with n vertic es taken fr om some non- trivial minor-close d family of gr aphs. F or every fixe d k , if G c ontains an induc e d cycle of size k , then it c an b e found in O ( n ) exp e cte d time. Pr o of. There is so me abso lute constant d , s o that G is d -degenera ted. Orie nt the edges so that the ma x im um o utdegree is at most d and denote the r esulting graph D = ( V , E ). W e now use the technique o f ra ndom separ ation. Ea ch v ertex v ∈ V of the graph is indep endently r e moved with pr obability 1 / 2, to g et some new directed graph D ′ . Now exa mine some (undirected) induced cycle o f size k in the o riginal directed graph D , and deno te its vertices b y U . The probability that a ll the vertices in U remained in the graph and all vertices in N + ( U ) − U were removed from the gr a ph is at least 2 − k ( d +1) . W e employ the color- co ding metho d to the g raph D ′ . Choose a random color- ing of the vertices of D ′ with the k colo rs { 1 , 2 , . . . , k } . F or each vertex v co lored i , if N + ( v ) contains a vertex with a color which is neither i − 1 nor i + 1 (mo d k ), then it is r emov ed fr om the g raph. F or ea ch induced cycle of size k , its vertices will receive distinct colo rs and it will remain in the graph with proba bilit y at least 2 k 1 − k . W e now use the O ( n ) time algor ithm fro m [4] to find a multicolored cycle of length k in the resulting graph. If such a c ycle exists, then it m ust b e an induced c y cle. Since k and d are constants, the a lgorithm succeeds with some small constant pr obability and the e xp e cted running time is as needed. ⊓ ⊔ The next theorem shows how to derando mize this alg orithm while incurring a loss of only O (log n ). Theorem 8. Supp ose that G is a gr aph with n vertic es taken fr om some non- trivial minor-close d family of gr aphs. F or every fixe d k , t her e is an O ( n lo g n ) time deterministic algorithm for finding an induc e d cycle of size k in G . Pr o of. Denote G = ( V , E ) and assume that G is d -degener ated. W e dera ndo mize the algor ithm in Theorem 7 using an ( n, dk + k )-family of p erfect hash functions. This is a fa mily of functions from [ n ] to [ dk + k ] such that fo r every S ⊆ [ n ], | S | = dk + k , there exists a function in the family that is 1-1 on S . Such a family of size e dk + k ( dk + k ) O (log ( dk + k )) log n can b e efficiently co nstructed [20]. W e think o f each function as a coloring of the vertices with the dk + k color s C = { 1 , 2 , . . . , dk + k } . F or every co mb ination of a color ing, a s ubset L ⊆ C of k colors and a bijection f : L → { 1 , 2 , . . . , k } the following is p erfo r med. All the vertices that got a co lor from c ∈ L now get the colo r f ( c ). The other vertices are removed fr om the gra ph. The vertices of the resulting g raph are colo red with the k colors { 1 , 2 , . . . , k } . Examine some induced cycle of size k in the origina l gr a ph, and deno te its v ertices by U . There exists some colo ring c in the family o f p er fect hash functions for which all the v ertices in U ∪ N + ( U ) r eceived different colors. Now le t L b e the k colors of the v ertices in the cycle U and let f : L → [ k ] be the bijection that giv es consecutive colors to vertices along the cycle. This mea ns that for this choice of c , L , and f , the induced cycle U will r emain in the gra ph as a multicolored cycle, whereas a ll the vertices in N + ( U ) − U will be remov ed fr o m the gra ph. W e pro ceed as in the previous algor ithm. B etter dependence on the param- eters d and k can b e obtained using the res ults in [3]. ⊓ ⊔ 5 Concluding R emarks – The algo r ithm for finding a dominating set in gra phs with an excluded mi- nor, prese nted in this paper , generaliz es a nd improves known a lg orithms for planar graphs and gra phs with b ounded genus. 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