A comparison of two approaches for polynomial time algorithms computing basic graph parameters

A comparison o f t w o approac hes f or p olynomia l time algorithms computing basic graph pa rameters ∗ † F rank Gurski ‡ Marc h 26, 2 018 Abstract In this pap er we compare a nd illustrate the a lgorithmic use o f g raphs of b ounded tree- width and gra phs of bo unded clique-width. F or this purp ose we g ive p olynomial time algorithms for computing the four bas ic graph pa r ameters indep endence num b er, clique nu mber , chromatic nu mber , and clique c overing n umber on a given tree str uctur e of gr aphs of bounded tree - width and graphs of bounded clique- width in p olynomial time. W e also present linear time algorithms for computing the latter four basic gr aph pa rameters on trees, i.e. graphs of tree-width 1 , and on co-g raphs, i.e. graphs of clique- width at most 2. Keyw ords: g raph algor ithms, g raph parameters , clique-width, NLC-width, tree-width 1 In tro du c tion A graph p arameter is a mapping that asso ciates ev ery graph with a p ositiv e intege r. W el l kno wn graph p arameters are indep endence n u mb er, dominating num b er, and chromatic num- b er. In general the computation of such parameters for s ome giv en graph is NP-hard . In th is w ork we giv e fi xed-parameter tractable (fp t) algorithms for computing basic graph parameters restricted to graph classes of b ound ed tree-width and graph classes of b oun d ed clique-width. The tree-width of graphs has b een defi ned in 1976 b y Halin [Hal76] and indep endently in 1986 b y Rob ertson and Seymour [RS86] by the existence of a tree decomp osition. In tuitive ly , the tree-width of some graph G measures h ow far G differs fr om a tree. Tw o more p o werful and more rece nt graph paramete rs are clique-width 1 and NLC-width 2 b oth d efined in 1994, by Courcelle and Olariu [CO00] and b y W ank e [W an94 ], resp ectiv ely . The clique-width of a graph G is the least in teger k suc h that G can b e defined b y op erations on ve rtex-lab eled graphs u s ing k lab els. These op erations are the v ertex disjoin t union, the addition of edges b et ween vertice s controlle d b y a label pair, and the relab eling of vertice s. The NLC-width of a graph G is defined similarly in terms of closely related op erations. The ∗ Small parts of this pap er h a ve b een published in an extended abstract [EGW01]. † This paper is a summary of t wo chapters of [Gur07]. ‡ Heinrich-Heine Universit¨ at D ¨ ussel dorf, D epartment of Computer S cience, D- 40225 D ¨ usseldorf , German y , E-Mail: gurski-corr@acs.uni-duesseldorf.de, 1 The operations in the definition of the graph parameter clique-width w ere first considered by Courcelle, Engelfriet, and R ozenberg in [CER91] and [CER93]. 2 The abb reviation NLC results from the node lab el con trolled em b edding mechanism originally defined for graph grammars [ER97 ]. 1 only essenti al difference b et ween th e comp osition mechanisms of clique-width b ounded graphs and NLC-w id th b oun ded graphs is the addition of edges. In an NLC-width comp osition the addition of edges is com bined with the u nion op eration. In tuitive ly , the clique-width and NLC-width of some graph G m easure h o w far G or its edge complemen t graph differs from a clique (i.e. a complete graph). See [BK07] and [HOSG07] for t wo recen t s u rv eys on tree-width and clique-width. One of the m ain reasons for regarding tree-width and clique-width is that a lot of hard problems b ecome solv able in p olynomial wh en restricted to graph classes of b oun d ed tree- width and graph classes of b oun ded clique-width . In this pap er we present tw o dynamic programming schemes to solv e graph problems on a giv en tree decomp osition (Chapter 3) and graph problems on a giv en clique-width expression (Chapter 4). These and similar dyn amic programming approac hes ha v e b een used in [Arn85],[AP89],[Bo d87],[Bo d88a], [Bod 90],[Hag00],[KZN00],[ZFN00],[INZ03] to solv e a large num b er of NP-complete graph problems on graph classes of b ounded tree-width and in [W an94 ], [EGW01], [GK03],[KR03],[T o d 03],[GW06], [MRA G06 ],[Rao06],[ST07],[Rao07], [Gur07] to solv e a large num b er of NP-complete graph problems on graph classes of b ounded clique-width. W e apply our t wo ap p roac hes in o rd er to compute the four basic graph parame- ters indep end ence n umber, clique num b er, c hromatic n umb er , and clique co ve rin g num b er on a giv en tree structure in p olynomial time. It is well known that th e compu tation of all four parameters is NP-complete on general graphs [GJ79]. The run ning time of our algorithms is exp onen tial in the tree-width or clique-width k but p olynomial in the instance size. Thus if we restrict our problems to graph classes of b ou n ded wid th s, parameter k will o ccur as a constan t in the run ning time and w e obtain p olynomial time paramet erized complexit y algorithms, see the b o oks [Nie06], [F G06], an d [DF99] for su rv eys. W e also pr esen t linear time algorithms f or compu ting the latter f ou r basic graph parameters on trees, i.e. graphs of tree-width 1, and on co-graphs, i.e. graphs of clique-width at most 2. Regarding theoretic ally resu lts f rom mon ad ic second order logic [CMR00, CM93], the existence o f the solutions for compu ting th e indep endence num b er and clique num b er on graphs o f b oun ded tree-width or graphs of b ounded c lique-width is kno wn. Nev ertheless our sho wn dyn amic p rogramming s olutions on a giv en tree str ucture are more feasible. Th e same remark holds true regarding complement problems on graphs of b oun ded clique-width. F urth er, this pap er compares the t wo main app roac hes whic h are used to solv e g rap h p roblems on tree-structured graph classes. Finally , in S ection 5 we discuss the v ertex co v er num b er and the dominating num b er as t w o further well kno wn graph parameters wh ic h can b e computed in p olynomial time o n graphs of b ound ed tree-width and graphs of b oun ded clique-width. F urth er we stress that b oth given dynamic programming approac hes to solv e p roblems along a tree d ecomp osition and along a clique-width expression are u seful. 2 Preliminaries 2.1 Definitions of graph parameters wit h algorithmic applications One of the most famous tree structured graph classes are graph s of b ounded tree-width. The notion of tree-width wa s d efined in the 1980s b y Rob ertson a nd Seymour in [RS86] as follo ws. 2 Definition 2.1 (TW k , tree-width, [RS86]) A tree decomp osition of a gr aph G = ( V G , E G ) is a p air ( X , T ) wher e T = ( V T , E T ) is a tr e e and X = { X u | u ∈ V T } is a family of subsets X u ⊆ V G , one for e ach no de u of T , such that the f ol lowing thr e e c ondition s hold true. • ∪ u ∈ V T X u = V G . • F or every e dge { v 1 , v 2 } ∈ E G , ther e is some no de u ∈ V T such tha t v 1 ∈ X u and v 2 ∈ X u . • F or eve ry vertex v ∈ V G the sub gr aph of T induc e d by the no des u ∈ V T with v ∈ X u is c onne cte d. The wid th of a tr e e de c omp osition ( X = { X u | u ∈ V T } , T = ( V T , E T )) is max u ∈ V T | X u | − 1 . The tr ee-width of a gr aph G (denote d by tr e e-width ( G ) ) is the smal lest inte ger k such that ther e is a tr e e de c omp osition ( X , T ) for G of width k . By TW k we denote the set of al l gr aph s of tr e e-width at most k . Fig. 1 shows a graph G and a tree decomp osition of width 2 for G . 1 u u 2 3 u u 5 6 u u 7 u 8 u 4 7 2 6 G 3 1 5 10 9 4 8 T 1 2 2 7 3 3 3 5 7 7 5 9 9 8 7 8 7 6 5 9 4 8 9 10 Figure 1: A graph G of tree-width 2 and a tree decomp osition ( X , T ) of width 2 for G . Next we giv e s ome examples for graph classes of b oun ded tree-width. T r ees hav e tree- width 1 [Bod98]. Series parallel graphs ha v e tree-width at most 2 [W C83]. Halin graphs ha ve tree-width at m ost 3 [Bo d88b ]. k -outerplanar graph s ha ve tree-width at most 3 k − 1 [Bod 88b]. A more d etailed o verview on graph classes of b oun ded tree-width can b e found in [Bod 86], [Bo d88b]. On th e other hand, the tree-width of complete graphs and th us of co-graphs (whic h are defined in Section 4.3 ) is not b ounded [Bo d 98]. Tw o more recen t p arameters are clique-width and NLC-width, wh ich are originally defined for lab eled graph s. Let [ k ] := { 1 , . . . , k } b e th e set of all in tegers b et we en 1 and k . W e work with fi n ite undirected lab eled gr aphs G = ( V G , E G , lab G ), w here V G is a finite set of v ertic es la b eled b y some mapping lab G : V G → [ k ] and E G ⊆ {{ u, v } | u, v ∈ V G , u 6 = v } is a finite set 3 of e dges . A lab eled graph J = ( V J , E J , lab J ) is a sub gr aph of G if V J ⊆ V G , E J ⊆ E G and lab J ( u ) = lab G ( u ) for all u ∈ V J . J is an induc e d sub gr aph of G if add itionally E J = { { u, v } ∈ E G | u, v ∈ V J } . The lab eled graph consisting of a single v ertex lab eled b y a ∈ [ k ] is denoted b y • a . The notion of clique-width of lab eled graphs is defined by Courcelle and Olariu in [CO00] as follo ws. Definition 2.2 (CW k , clique-w idth, [CO00]) L et k b e some p ositive i nte ger. The class CW k of lab e le d gr aphs is r e cu rsively define d as fol lows. • The single vertex • a lab ele d by some a ∈ [ k ] is in CW k . • L et G ∈ CW k and J ∈ CW k b e two vertex disjoint lab ele d gr aphs. Then G ⊕ J := ( V ′ , E ′ , lab ′ ) define d by V ′ := V G ∪ V J , E ′ := E G ∪ E J , and lab ′ ( u ) :=  lab G ( u ) if u ∈ V G lab J ( u ) if u ∈ V J , for all u ∈ V ′ is in CW k . • L et a, b ∈ [ k ] b e two distinct i nte gers and G ∈ CW k b e a lab ele d gr aph then – ρ a → b ( G ) := ( V G , E G , lab ′ ) define d by lab ′ ( u ) :=  lab G ( u ) if lab G ( u ) 6 = a b if lab G ( u ) = a , for all u ∈ V G is in CW k and – η a,b ( G ) := ( V G , E ′ , lab G ) define d by E ′ := E G ∪ {{ u, v } | u, v ∈ V G , u 6 = v , lab ( u ) = a, lab ( v ) = b } is in CW k . The clique-width of a lab e le d gr aph G (denote d by clique- width ( G ) ) is the le ast inte ger k such that G ∈ CW k . The notion of NLC-wid th of lab eled graph s is defined by W ank e in [W an 94 ] as follo ws. Definition 2.3 (NLC k , NLC-width, [W an94]) L e t k b e some p ositive inte ger. The gr aph class NLC k of lab e le d gr aphs is r e cu rsively define d as fol lows. 1. The single vertex gr aph • a for some a ∈ [ k ] is in NLC k . 2. L et G = ( V G , E G , lab G ) ∈ NLC k and J = ( V J , E J , lab J ) ∈ NLC k b e two vertex disjoint lab ele d gr aphs and S ⊆ [ k ] 2 b e a r elation, then G × S J := ( V ′ , E ′ , lab ′ ) define d by V ′ := V G ∪ V J , E ′ := E G ∪ E J ∪ {{ u, v } | u ∈ V G , v ∈ V J , ( lab G ( u ) , lab J ( v )) ∈ S } , and lab ′ ( u ) :=  lab G ( u ) if u ∈ V G lab J ( u ) if u ∈ V J , ∀ u ∈ V ′ is in NLC k . 4 3. L et G = ( V G , E G , lab G ) ∈ NLC k b e a lab ele d g r aph and R : [ k ] → [ k ] b e a function, then ◦ R ( G ) := ( V G , E G , lab ′ ) define d by lab ′ ( u ) := R ( lab G ( u )) , ∀ u ∈ V G is in NLC k . The NLC-width of a lab ele d gr aph G (denote d by NLC-width ( G ) ) is the le ast inte ge r k su ch that G ∈ NLC k . An expression built with the op er ations • a , ⊕ , ρ a → b , η a,b for integ ers a, b ∈ [ k ] is called a clique-width k -expr ession . An expr ession X built with the op erations • a , × S , ◦ R for a ∈ [ k ], S ⊆ [ k ] 2 , and R : [ k ] → [ k ] is called an NLC-width k -expr ession . The clique- width (the NLC-width ) of an un lab eled graph G = ( V , E ) is the smallest intege r k , such that there is some m apping lab : V → [ k ] suc h t h at the labeled graph ( V , E , lab ) has clique-width at most k (NLC-width at m ost k , resp ectiv ely). The graph defi ned by expression X is denoted b y v al( X ). By th e definition of k -expressions it is e asy to verify that graphs of b ound ed clique-width and graphs of b oun ded NLC-width are closed und er taking ind uced subgraph s. Ev ery clique-width k -expr ession X has by its r ecursiv e defin ition a tree structure that is called the clique-width k -expr ession-tr e e T f or X . T is an ordered ro oted tree whose lea ves corresp ond to the v ertices of graph v al( X ) and the inner no des corresp ond to the op erations of X , see [EGW03]. In the same wa y we d efine the NLC-width k -expression-tree for ev ery NLC-width k -expr ession, see [GW00]. If integ er k is kno wn fr om the con text or irrelev ant for the discus s ion, then w e sometimes use th e simplified notion expr ession-tr e e for th e n otion k -expression-tree. The follo wing example sho ws th at ev ery clique K n , n ≥ 1, has clique-width 2 and NLC - width 1 and that every path P n has clique-width at m ost 3 and NLC-width at most 3. Example 2.4 (1.) Every clique K n = ( { v 1 , . . . , v n } , {{ v i , v j } | 1 ≤ i < j ≤ n } ) , n ≥ 2 , has clique-width 2, by the fol lowing r e c ursively define d expr essions X K n . X K 2 := η 1 , 2 ( • 1 ⊕ • 2 ) X K n := η 1 , 2 ( ρ 2 → 1 ( X K n − 1 ) ⊕ • 2 ) , if n ≥ 3 (2.) Every p ath P n = ( { v 1 , . . . , v n } , {{ v 1 , v 2 } , . . . , { v n − 1 , v n }} ) has clique- width at most 3, by the fol lowing r e c ursively define d expr essions X P n . X P 3 := η 2 , 3 ( η 1 , 2 ( • 1 ⊕ • 2 ) ⊕ • 3 ) X P n := η 2 , 3 ( ρ 3 → 2 ( ρ 2 → 1 ( X P n − 1 )) ⊕ • 3 ) , if n ≥ 4 (3.) Every clique K n , n ≥ 1 , has NLC-width 1, by the fol lowing r e cu rsive ly define d expr es- sions X K n . X K 1 := • 1 X K n := X K n − 1 × { (1 , 1) } • 1 , if n ≥ 2 (4.) Every p ath P n has NLC-width at most 3, by the fol lowing r e cursively define d expr essions X P n . X P 3 := ( • 1 × { (1 , 2) } • 2 ) × { (2 , 3) } • 3 X P n := ◦ { (1 , 1) , (2 , 1) , (3 , 2) } ( X P n − 1 ) × { (2 , 3) } • 3 , if n ≥ 4 5 Next we giv e some examples for graph classes of b oun ded clique-width. Di stance h eredi- tary graph s hav e clique-width at most 3 [GR00]. Co-graphs, i.e. P 4 -free graph s hav e clique- width at most 2 [CO 00]. F urther, man y graph classes defined b y a limited num b er of P 4 ha ve b ound ed clique-width, e.g. P 4 -reducible graph s, P 4 -sparse graph s, P 4 -tidy , and ( q , t )- graphs [CMR00, MR99]. A recen t sur vey on graph classes of b ounded clique-width is giv en in [KLM07]. On the other hand, the clique-width of p erm u tation graphs, interv al graph s, grids and planar graphs is not b oun d ed [GR00]. 2.2 Relations b et ween graph parameters Next w e briefly sur v ey the relation b et wee n tree-width, clique-width , and NLC-width. Theorem 2.5 ( [Joh98 ]) Every gr aph of clique-width k has NLC- width at most k , and every gr aph of NLC- width at most k has clique-width at most 2 k . Th u s w e conclude that a set of graphs has b ounded clique-width if and only if it h as b ound ed NLC-width. Both concepts are useful, because it is sometimes muc h more com- fortable to u se NLC-width expr essions instead of clique-width expressions and vice versa, resp ectiv ely , see Chapter 4. It is well kno wn that ev ery graph of b ounded tree-width also has b oun d ed clique-width, see [CO00, C R05, W an94 ]. The b est kno wn b ound is the follo wing one sho wn by Corneil and Rotics. Theorem 2.6 ( [C R05]) L et G b e a gr aph of tr e e- width k , then G has cliqu e-width at most 3 · 2 k − 1 . Con versely , the tree-width of a graph can not b e b ound ed in its clique-width in general. This sho ws e.g. the set of all complete graph s ( K n has clique-width 2 and tree-width n − 1). Under the add itional assumption that w e restrict to graphs that do not con tain arbitrary large complete bipartite graph s K n,n , th e tree-width of a graph can b e b ounded in its clique- width [GW00 ]. Thus, if w e restrict to graphs of b ounded v ertex d egree or planar graph s, a set of graphs has b ounded tree- wid th or b ound ed path-wid th if and only if it has b ounded clique-width or b ounded linear clique-width, r esp ectiv ely . A fur ther v ery useful and in teresting relation b et we en tree-width and clique-width h as b een shown using the concept of line graph s. A set of graphs has b ounded tree-width if and only if the corresp onding set of line graph s has b ounded clique-width [GW07]. 2.3 Definitions of basic graph parameters Next w e giv e definitions for the four basic graph parameters ind ep endence num b er, clique n umb er, c hromatic num b er , and clique co v ering num b er. Problem 2.7 ( Indep endent Set, [GT20] in [GJ79]) Instance: A gr aph G = ( V G , E G ) and a p ositive inte ger s ≤ | V G | . Question: Is ther e an indep endent set of size at le ast s in G , i.e. a subset V ′ ⊆ V G , such that | V ′ | ≥ s and no two vertic es of V ′ ar e joine d by an e dge in E G ? 6 The maxim u m v alue s s uc h that G has an ind ep endent set of size s is denoted as the indep endenc e nu mb er of graph G , den oted by α ( G ). Problem 2.8 ( Clique, [GT19] in [GJ79]) Instance: A gr aph G = ( V G , E G ) and a p ositive inte ger s ≤ | V G | . Question: Is ther e a cliqu e of size at le ast s in G , i.e. a subset V ′ ⊆ V G , such that | V ′ | ≥ s and every two v e rtic es of V ′ ar e joine d by an e dge in E G ? The maximum v alue s su c h that G has a clique of size s is denoted as th e clique numb er of G , denoted by ω ( G ). Problem 2.9 ( P artition In to Indep enden t Sets, [GT4] in [GJ79]) Instance: A gr aph G = ( V G , E G ) and a p ositive inte ger s ≤ | V G | . Question: Is ther e a p artition of V G into s disjoint sets V 1 , . . . , V s such tha t V 1 ∪ · · · ∪ V s = V G and no set V t , 1 ≤ t ≤ s , ha s two adjac ent vertic es? The minim um v alue s suc h that G has a partition into s indep end en t sets is denoted as the chr omatic numb er of G , denoted by χ ( G ). Equiv alen tly and motiv ating the notation c hromatic num b er, χ ( G ) is the least in teger s , suc h that there is a vertex c oloring col : V G → { 1 , . . . , s } suc h th at for eve ry pair of adjacen t vertic es v 1 , v 2 ∈ V G , v 1 6 = v 2 , it holds col( v 1 ) 6 = col( v 2 ). Problem 2.10 (P artition In to Cliques, [GT15] in [GJ79]) Instance: A gr aph G = ( V G , E G ) and a p ositive inte ger s ≤ | V G | . Question: Is ther e a p artition of V G into s disjoint sets V 1 , . . . , V s such tha t V 1 ∪ · · · ∪ V s = V G and every set V t , 1 ≤ t ≤ s , induc es a c omplete sub gr aph? The minimum v alue s suc h th at G has a partition into s cliques is d enoted as the clique c overing numb er of G , denoted b y θ ( G ). The graph parameters α , ω , χ , and θ pla y an imp ortan t rule in the fi eld of the researc h of p erfect graph s. One of the most famous c haracteriza tions for these graph s is that a graph G is p erfect if and only if for ev ery induced sub graph H of G it holds ω ( H ) = χ ( H ), if and only if for ev ery induced subgraph H of G it holds α ( H ) = θ ( H ). Examp les for perfect graph classes are b ip artite graphs, c hordal graph s, an d co-graphs, see [Hou06] for an o v erview. A furth er c haracterization for per f ect graphs is that a graph G is p erfect if and only if G con tains no C 2 n +1 and no C 2 n +1 as an induced sub graph. Since the cycle on 5 vertice s h as tree-width 2 and clique-width 3, w e conclude that graphs of tree-width at most k and graphs of cli qu e-width at most k are not p erfect for ev ery intege r k ≥ 2 and k ≥ 3, resp ectiv ely . In T able 1 we sur v ey the r esults of this pap er. 7 ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ ❵ graph class parameter α ω χ θ trees/forest Lin Thm 3.7 Lin folk Lin folk Lin Thm 3.8 TW k Lin Thm 3.3 Lin Thm 3.4 P Thm 3.5 P Thm 3.6 co-graphs Lin Thm 4.7 Lin Thm 4.8 Lin Th m 4.9 Lin T h m 4.10 CW k Lin Thm 4.3 Lin Thm 4.4 P Thm 4.5 P Thm 4.6 T able 1: Ove r v iew on the time complexit y of computing α , ω , χ , and θ o n sp ecial graph classes. 3 T ree-width and p olynomial time algorithms 3.1 A general framew ork In ord er to solv e hard p roblems r estricted to graph classes of b ound ed tree-width, w e w ill p erform a d y n amic programming sc heme on the tree d ecomp osition int ro duced in Definition 2.1. Although compu ting the tree-width of a giv en graph is NP-complete [A CP87], for ev ery fixed intege r k , the problem to decide whether a giv en graph G has tree-width at most k can b e solv ed in linear time and in the case of a p ositiv e answer a tree decomp osition of width k for G can b e found in the same time [Bo d96]. F or the pu rp ose of conv en ience we wa nt to restrict our algorithms to sp ecial b inary decomp ositions wh ic h is alwa ys p ossible b y th e follo wing theorem. Theorem 3.1 ( [K lo94 ]) L et G b e a gr aph of tr e e-width k . Then G has a tr e e de c omp osition ( X = { X u | u ∈ V T } , T = ( V T , E T )) of width k , such that a r o ot r of T c an b e chosen such that the fol lowing five c onditions ar e fulfil le d. 1. Every no de of T has at most two childr en. 2. If a no de u of T has two childr en v and w , then X u = X v = X w . In this c ase u is c al le d a join n o de . 3. If a no de u of T has one child v , then one of the fol lowing tow c onditions hold true. (a) | X u | = | X v | + 1 and X v ⊂ X u . In this c ase u is c al le d an introdu ce no de . (b) | X u | = | X v | − 1 and X u ⊂ X v . In this c ase u is c al le d a forget no de . 4. If a no de u is a le af of T , then | X u | = 1 . 5. | V T | ∈ O ( k · | V G | ) . A tree decomp osition whic h fulfills the five conditions of Theorem 3.1 is called a nic e tr e e de c omp osition and can b e foun d in linear time [Klo94 ]. Let G b e a graph of tree-width k and ( X = { X u | u ∈ V T } , T = ( V T , E T )) tree decomp osition with ro ot r for G . F or some no de u of T we define T u as the subtree of T ro oted at u and b y X u the set of all X v , v ∈ V T u . F urth er b y G u w e define the subgraph of G wh ic h is defined by all n o des in sets X v where v = u or v is a child of u in T , i.e. G u is d efined by tree decomp osition ( X u , T u ). The sets X u ∈ X will b e denoted as b ags . 8 Our solutions are b ased on a sep ar ator pr op erty of the vertic es of graphs giv en b y a tree decomp osition ( X , T ) of w idth at most k . Let u , w b e tw o no des of T an d v 1 ∈ X u , v 2 ∈ X w t w o v ertices of the graph G defined b y d ecomp osition ( X , T ). If th ere exists some no de s of T on the path from u to w in T suc h that v 1 6∈ X s and v 2 6∈ X s , th en v 1 and v 2 are not adjacen t in G , see Fig. 2. Thus the at most k + 1 v ertices of bag X s separate the vertic es in bags b elo w X s from the remaining vertice s of G . 2 v v 1 v 1 2 v u w s T Figure 2: Separator prop erty of some graph G defined by tree deco mp ositions ( X , T ). Let u, w ∈ V T and v 1 ∈ X u , v 2 ∈ X w b e t wo vertice s of graph G . If v 1 6∈ X s and v 2 6∈ X s , w e kno w that vertices v 1 and v 2 are not adjacen t in G . In order to solv e graph problems on tree-width b ounded graphs we will use the follo wing b ottom up dyn amic programming sc heme. Theorem 3.2 L e t Π b e a gr aph pr oblem and k b e a p ositive inte ger. If ther e is a mapping F that maps eve ry tr e e-dec omp osition ( X = { X u | u ∈ V T } , T = ( V T , E T )) with r o ot r of width k onto some structur e F ( r ) , such that for al l no des v , w of T , 1. the size of F ( v ) is p olynomial ly b ounde d in the size of ( X v , T v ) , 2. the answer to Π f or G v is c omputable in p olynomial time fr om F ( v ) , 3. for every le af u of T structur e F ( u ) is c omputable in time O (1) , 4. for eve ry join no de u with childr en v , w structur e F ( u ) is c omputable in p olynomial time fr om F ( v ) and F ( w ) , and 5. for every intr o duc e no de and every for get no de u with child v structur e F ( u ) is c om- putable in p olynomia l time fr om F ( v ) . Then for every de c omp osition ( X , T ) of width k , the answer to Π for gr aph G r is c omputable in p olynom ial time fr om de c omp osition ( X , T ) . There are fu r ther d ynamic programming app r oac hes to solve h ard p roblems on tree-width b ound ed graph s. F or example in [AP89 ] the p erfect elimination order of the vertic es of a partial k -tree is used to solve hard problems on tree-width b oun ded graph s. 9 3.2 Computing α , ω , χ , and θ on gr aphs of b ounded t r ee-width W e next app ly the general sc h eme of Theorem 3.2 for computing the four basic graph param- eters α , ω , χ , and θ on graph s of b ounded tree-width in p olynomial time. 3.2.1 Indep endence n umber First we consider the problem of find in g the size of a maxim um ind ep endent set (Problem 2.7) in a graph give n by some tree decomp osition. Let ( X = { X u | u ∈ V T } , T = ( V T , E T )) b e a tree d ecomp osition for some graph G of width k w ith ro ot r . F or ev ery nod e u of T w e define a 2 k +1 -tuple F ( u ) which contai n s f or ev ery subset X ⊆ X u an integer a X , i.e. F ( u ) = ( a X | X ⊆ X u ). Th e v alue of a X denotes the s ize of a maxim u m indep endent set U ⊆ V G u in graph G u suc h that U ∩ X u = X . Note that b ecause of our separator prop ert y , ve rtices from U − X will not get an y further edges in bag X u or some bag X w for some no d e w which is n ot a c hild of u in T . Then F ( r ) is b oun ded in k indep endentl y of the size of ( X , T ), b ecause by the definition ev ery bag con tains at most k + 1 v ertices and thus F ( r ) h as at most 2 k +1 en tries. T he follo wing observ atio n s sho w th at for ev ery leaf u of T structure F ( u ) is computable in time O (2 k +1 ), for eve ry join nod e u with c hildren v , w structure F ( u ) is computable in time O (2 k +1 ) from F ( v ) and F ( w ), and for ev ery introdu ce n o de an d ev ery forget n o de u w ith c hild v stru cture F ( u ) is computable in time O (2 k +1 ) from F ( v ). 1. If u is a leaf of T , su c h that X u = { v 1 } for some v 1 ∈ V G . W e defin e F ( u ) := ( a ∅ = 0 , a { v 1 } = 1 ). 2. If u is a join n o de with c hildren v , w of T . Let F ( v ) = ( a X | X ⊆ X v ), F ( w ) = ( b X | X ⊆ X w ), and i X b e the size of the largest indep endent set in X ⊆ X u . T hen we define F ( u ) := ( c X | X ⊆ X u ), where ∀ X ⊆ X u , c X := a X + b X − i X . 3. If u is an in tro du ce no de with c hild v of T , su c h that X u − X v = { v ′ } for some v ′ ∈ V G . Let F ( v ) = ( a X | X ⊆ X v ). T hen we define F ( u ) = ( b X | X ⊆ X u ), wh ere ∀ X ⊆ X v , b X := a X and b X ∪{ v ′ } :=  a X + 1 if v ′ is not adj acen t to some v ertex from X −∞ else 4. If u is a f orget no d e with c hild v of T , s u c h that X v − X u = { v ′ } for some v ′ ∈ V G . Let F ( v ) = ( a X | X ⊆ X v ). Then w e define F ( u ) := ( b X | X ⊆ X u ), where ∀ X ⊆ X u , b X := max { a X , a X ∪{ v ′ } } . After a dynamic p rogramming computation of F ( r ) w e can easily compute the size of a maxim um ind ep endent set in graph G by α ( G ) := max a ∈ F ( r ) a Theorem 3.3 The indep endenc e numb er of a gr aph of b ounde d tr e e-width c an b e c ompute d in line ar time. In [Chl02] it is shown th at for ev ery graph G the v alue of | V G | − tree-width( G ) alwa ys is an upp er b ound for the indep endence n umb er α ( G ). 10 3.2.2 Clique n um b er Next w e consider the pr oblem of findin g the size of a maximum clique (Problem 2.8 ) in a graph giv en b y some tree decomp osition. Let G b e some graph and ( X = { X u | u ∈ V T } , T = ( V T , E T )) b e a tree d ecomp osition of width k for G . In order to compute the v alue of ω ( G ) obviously a similar solution as given for indep enden t set problem ab ov e is p ossible. Alternativ ely one could use the well kno wn result that for eve ry clique C = ( V C , E C ) in graph G there exists s ome bag X u ∈ X such that V C ⊆ X u , see [BM93]. This allo w s us to compute the v alue of ω ( G ) for graph G of tree-width k by ω ( G ) := max u ∈ V T max C ⊆ X u G [ C ] cl ique | C | . Theorem 3.4 The clique numb er of a gr aph of b ounde d tr e e- width c an b e c ompute d in line ar time. 3.2.3 Chromatic n um b er F urth er w e consider the pr oblem of fi nding the minimum num b er of ind ep endent sets (Problem 2.9) co v ering a graph giv en b y some tree decomp osition. Let ( X = { X u | u ∈ V T } , T = ( V T , E T )) b e a tree decomp osition for some graph G of width k with r o ot r . F or ev ery nod e u of T , we d efine a set F ( u ) whic h con tains for ev ery partition of V G u in to indep enden t set s V 1 , . . . , V s a 2 k +1 -tuple t = ( . . . , a X , . . . , a ) which con tains for ev ery nonempty subset X ⊆ X u a b o olean v alue a X and one in teger a . F or some disjoin t partition V 1 , . . . , V s of V G u in to indep enden t sets, the v alue of a X is 1, if and only if V i ∩ X u = X for some 1 ≤ i ≤ r and the v alue of a denotes the num b er of ind ep endent sets V i suc h that V i ∩ X u = ∅ . Then F ( r ) is p olynomially b oun ded in the size of ( X , T ), b ecause eve r y elemen t of F ( r ) has 2 k +1 en tries, 2 k +1 − 1 from { 0 , 1 } and one from { 1 , . . . , | V G |} , i.e. | F ( r ) | ≤ 2 2 k +1 − 1 · | V G | . The follo win g observ atio n s sho w that for eve ry leaf u of T structur e F ( u ) is co mp utable in time O (1), for ev ery j oin nod e u with children v , w structure F ( u ) is computable in p olynomial time from F ( v ) and F ( w ), and for ev ery introd uce no d e and ev ery forget n o de u w ith c h ild v s tr ucture F ( u ) is compu table in p olynomial time from F ( v ). 1. If u is a leaf of T , s uc h that X u = { v 1 } for some v 1 ∈ V G . W e define F ( u ) := { ( a { v 1 } = 1 , a = 0) } . 2. If u is a join no de with c hildren v , w of T . Then we define F ( u ) := { ( . . . , a X , . . . , a + b ) | ( . . . , a X , . . . , a ) ∈ F ( v ) , ( . . . , b X , . . . , b ) ∈ F ( w ) , a X = b X , for all X ⊆ X u } . 3. If u is an in tro duce no de with c hild v of T , su c h that X u − X v = { v ′ } for some v ′ ∈ V G . W e consider every p artition of X v in to indep endent sets in order to insert a n ew indep end en t s et which just con tains v ertex v ′ and to extend one existing indep endent set of a partition of X v b y vertex v ′ . Th u s w e define F ( u ) as follo ws. F or ev ery tup le t ∈ F ( v ) w e insert a tu ple t ′ in to F ( u ) whic h con tains the same v alues as t and ad d itionally a { v ′ } := 1 . F urther for ev ery tuple t ∈ F ( v ) and ev ery X ⊆ X v suc h that a X = 1 in F ( v ) and X ∪ { v ′ } is an ind ep endent set of graph G , w e insert a tuple t ′ in to F ( u ) w hic h co ntains the same v alues as t but 11 a X := 0 and additionally a X ∪{ v ′ } := 1 . In b oth cases the v alue of a of t ′ is the s ame as in t . 4. If u is a f orget no d e with c hild v of T , s u c h that X v − X u = { v ′ } for some v ′ ∈ V G . W e kn o w that in ev ery p artition of X v there is exactly one indep enden t set X wh ic h con tains v ′ . If X d o es not con tain any fu r ther v ertex, we hav e to increase the v alue of a in F ( u ) by one, otherwise we kno w that X − { v ′ } is an indep end ent set in G u , i.e. a X −{ v ′ } = 1 in F ( u ). Th u s w e defin e F ( u ) as f ollo ws . F or eve ry tuple t = ( . . . , a X , . . . , a ) ∈ F ( v ) we insert a tuple t ′ := ( . . . , b X , . . . , b ) in to F ( u ) whic h is defin ed as follo ws. If a { v ′ } = 1, then w e define for every X ⊆ X v b X −{ v ′ } := a X and b := a + 1. O therwise (i.e. a { v ′ } = 0), w e define for eve ry X ⊆ X v b X −{ v ′ } := a X and b := a . Note that in all four cases set F ( u ) has at most 2 | X u | en tries. After a dynamic programming computation of F ( r ) w e can compute the chromati c num b er of graph G by χ ( G ) := min t ∈ F ( r ) P a ∈ t a . Theorem 3.5 The chr omatic numb er of a gr aph of b ounde d tr e e- width c an b e c ompute d in p olynomial time. In [Chl02] it is shown that for ev ery graph G th e v alue of tree-width( G ) + 1 is alw ays an upp er b ound for the c hromatic n u m b er χ ( G ). If we co nsid er Pr oblem 2 .9 for th e c ase that w e lo ok for a partition in to a minimum n umb er of in dep end ent edge sets, w e obtain the graph p arameter chromatic index, see [Viz64], [Gup66], whic h h as b een sh o wn to b e computable in linear time on graphs of b oun ded tree- width in [ZFN00]. 3.2.4 Clique co v ering n umber F urth er we consider the problem of finding the minimum num b er of cliques (Pr ob lem 2.10) co v ering a graph giv en by some tree decomp osition. Let ( X = { X u | u ∈ V T } , T = ( V T , E T )) b e a tree decomp osition for some graph G of width k w ith ro ot r . W e will pro ceed similarly to the solution giv en in Section 3.2.3 for computing the c h r omatic n umber. F or ev ery no de u of T , w e define a set F ( u ) whic h co ntains for ev ery disjoint partition of V G u in to cliques V 1 , . . . , V s a 2 k +1 -tuple t = ( . . . , a X , . . . , a ) whic h contai n s for every nonempty subset X ⊆ X u a b o olean v alue a X and one intege r a . F or some disj oin t partition V 1 , . . . , V s of V G u in to cliques, th e v alue of a X is 1, if and only if V i = X f or some 1 ≤ i ≤ r and the v alue of a denotes th e n umb er of cliques V i suc h th at V i ∩ X u = ∅ . Then F ( r ) is p olynomially b ounded in the size of ( X , T ), b ecause | F ( r ) | ≤ 2 2 k +1 − 1 · | V G | . F urth er for ev ery leaf u of T structure F ( u ) is computable in time O (1), for ev ery join no de u with c hildren v , w s tr ucture F ( u ) is computable in p olynomial time from F ( v ) and F ( w ), and for ev ery introduce no d e and every forget no d e u with c hild v stru cture F ( u ) is computable in p olynomial time from F ( v ). This follo ws b y step (1) an d (2) giv en in Section 3.2.3 and b y replacing indep endent set by clique in step (3) and (4) giv en in Section 3.2.3. After a dynamic programming computation of F ( r ) we can compute the clique co v ering n umb er of graph G by θ ( G ) := min t ∈ F ( r ) P a ∈ t a 12 Theorem 3.6 The clique c overing numb er of a gr aph of b ounde d tr e e- width c an b e c ompute d in p olynom ial time. 3.3 Computing α , ω , χ , and θ on t rees W e next show that for graphs of tree-width one, i.e. for trees, our shown algorithms can b e simplified. 3.3.1 Indep endence n umber Let T b e some tree. T he ind ep enden ce num b er α ( T ) can b e computed by α ( T ) = α ( T r ), where T r is th e corresp ond ing ro oted tree by c ho osing an arb itrary v ertex r of T as a ro ot. The v alue of α ( T r ) (and thus α ( T )) can b e computed b y d ynamic pr ogramming as follo ws. 1. If | V T r | = 1, i.e. r is a leaf of T r , then α ( T r ) := 1. 2. If | V T r | ≥ 2, i.e. r is an inner no de of T r , then α ( T r ) := max { X c c hild of r α ( T c ) , 1 + X g grand c hild of r α ( T g ) } Theorem 3.7 F or every tr e e its i ndep endenc e numb er c an b e c ompute d in line ar time. 3.3.2 Clique n um b er Ob viously f or ev ery tree T , if | V T | = 1, then ω ( T ) := 1 and if | V T | ≥ 2, then ω ( T ) := 2. 3.3.3 Chromatic n um b er Ob viously f or ev ery tree T , if | V T | = 1, then χ ( T ) := 1 and if | V T | ≥ 2, then χ ( T ) := 2, since ev ery tree is a bipartite graph . 3.3.4 Clique co v ering n umber Since trees are p er f ect, we k n o w that for eve ry induced subgraph H of some tree T it holds θ ( H ) = α ( H ) , and th us we can compute clique co v ering num b er θ ( T ) by the same algorithm as shown for α ( T ) ab o v e. Theorem 3.8 F or every tr e e its c liq ue c overing numb er c an b e c ompute d in line ar time. 3.4 Complemen t problems Let Π b e a decision problem for graphs. W e defin e the corresp ond ing c omplement pr oblem Π b y Π := { G | G satiesfies Π } . F or sev eral graph problems the co r resp ond in g complement problem is also of in terest. F or example the complemen t p roblem of the indep enden t set problem (Problem 2.7 ) is the clique problem (Problem 2.8) and the co mp lemen t prob lem of the partition in to ind ep endent sets problem (Problem 2.9) is the partition into cliques problem (Problem 2.10). 13 Since for some set of graphs L ⊆ TW k , the corresp onding set of complemen t graphs L := { G | G ∈ L} not necessarily has b ounded tree-width, the s olv abilit y of complement problems on tree-width b ounded graphs are worth while to researc h. In [GKS00] Gupta et al. giv e a logical framew ork for s olving complemen t problems on tree-width b ounded graphs in p olynomial time. 3.5 T ree-width and monadic second order logic On graph classes of b oun d ed tr ee-width, all graph prop erties and optimization problems whic h are expr essib le in monadic s econd ord er logic with quantificat ions ov er v ertices, v ertex sets, edges, and edge sets (MSO 2 -logic) are d ecidable in linear time [CM93]. This imp lies the existence of linear time algorithms for compu tin g the indep endence n umb er and clique n umb er on graphs of b oun ded tree-width. Note th at the problems partition into indep endent sets and partition in to cliques are not expressible in MSO 2 -logic. 4 Clique-width and p olynomial time algorithms 4.1 A general framew ork In order to solve hard problems restricted to graph classes of b ounded clique-width, w e recall a dy n amic programming approac h on th e tree structure of a clique-width or an NLC-width expression from [EGW01]. Recen tly it has b een s h o wn that co mp uting the clique-width and NLC-width of a giv en graph is NP-hard [FRRS06, GW05]. F or ev ery fixed intege r k ≤ 3 or k ≤ 2, the pr oblem to decide wh ether a giv en graph has clique-width at most k or NLC -w id th at most k can b e solv ed in p olynomial time and in th e case of a p ositiv e ans wer a k -expression can b e constructed in the same time [CPS85, CHL + 00, LdMR07]. F or ev ery fi xed in teger k ≥ 4 or k ≥ 3, the problem to decide whether a giv en graph has clique-width at most k or NLC -width at most k is still op en. Nev ertheless we can use the appro ximations for rank-width shown by Oum and Seymour in [OS06 , Oum05, Oum06] in order to obtain approxi mations for clique-width and a corresp ond ing clique-width expression. The b est kno wn resu lt is the follo wing. Theorem 4.1 ( [O um06]) F or every fixe d i nte ger k ther e is a O ( | V G | 3 ) algorithm that either outputs a clique - width (8 k − 1) -expr ession of an input gr aph G , or c onfirms that the clique- width of G is lar ger that k . Ev ery clique-width k -expr ession can be transformed in to an equiv alent NLC-width k - expression w ithin linear time [Joh98]. Th us, t h e last theorem implies that for ev ery fixed in teger k , for ev ery set L ⊆ CW k and ev ery set L ⊆ NLC k , we can assu m e within cubic time ev ery graph G ∈ L to b e giv en with some (8 k − 1)-expression. F or some no de u of expr ession-tree T , let T ( u ) b e the su btree of T ro oted at u . Note that tree T ( u ) is alwa ys an expression-tree. T he expression X ( u ) defi n ed by T ( u ) can simp ly b e determined by trav ersing the tree T ( u ) starting f rom the ro ot, where the left c hildr en are visited first. X ( u ) defines a (p ossibly) relab eled induced su bgraph G [ u ] of G . Our solutions are b ased on a neighb ourho o d pr op erty of the v ertices of grap h s given by a k -expression X defining a corresp ond ing k -expression tree T . F or every n o de u of T , the v ertices of subgraph G [ u ] form a k -mo du le, i.e. ev ery set V i = { lab( v ) = i | v ∈ V G [ u ] } , 14 1 ≤ i ≤ k , is a mo d u le of G [ V G − ( V 1 ∪ . . . ∪ V k ) ∪ V i ]. T hat is, all v ertices in set V i , 1 ≤ i ≤ k , will b e treated equally b y all op erations in T on the p ath from u to the r o ot of T , see Fig. 3. V 1 V 2 V 3 1 1 3 2 2 3 V 1 V 2 1 1 2 2 T u Figure 3: Neighbourh o o d prop ert y of graph s defined by k -expr ession tree T . F or ev ery n o de u ∈ V T , th e v ertices of subgraph G [ u ] can b e divided with r esp ect to their lab els into at most k mo du les V 1 , . . . , V k in the graph defin ed b y T . The tree structure of such k -expr essions can be used to solve hard prob lems by the fol- lo wing general b ottom up d ynamic programming sc heme. Theorem 4.2 ( [E GW01]) L et Π b e a gr aph pr oblem and k b e a p ositive inte ger. If ther e is a mapping F that maps e ach clique-width k -expr ession X onto some structur e F ( X ) , su ch that for al l clique -width k -expr essions X , Y and al l a, b ∈ [ k ] 1. the size of F ( X ) i s p olyno mial ly b ounde d in the si ze of X , 2. the answer to Π f or val ( X ) is c omputable in p olynomial time fr om F ( X ) , 3. F ( • a ) is c omputable in time O (1) , 4. F ( X ⊕ Y ) is c omputable in p olynomial time f r om F ( X ) and F ( Y ) , and 5. F ( η a,b ( X )) and F ( ρ a → b ( X )) ar e c omputable in p olyno mial time fr om F ( X ) . Then for every clique-width k - e xpr ession X , the answer to Π for gr aph val ( X ) is c omputable in p olynom ial time fr om expr ession X . Theorem 4.2 also works for NLC-width k -expressions built with the op erations • a , × S , and ◦ R instead of • a , ⊕ , η a,b , and ρ a → b . In this case F ( X × S Y ) has to b e computable in p olynomial time f r om F ( X ) and F ( Y ), and ◦ R ( X ) has to b e computable in p olynomial time from F ( X ). The giv en d ynamic programming app r oac h has b een used in [EGW0 1 ],[GW06], [Gur07], [GK03],[KR03],[Rao07] to solv e a large num b er of NP-co mp lete graph problems on grap h classes of b ounded clique-width. 4.2 Computing α , ω , χ , and θ on gr aphs of b ounded clique-width W e n ext give p olynomial time algorithms usin g f or computin g the four basic graph p arameters α , ω , χ , and θ on graphs of b ounded clique-width u sing the general sc heme of Theorem 4.2. F or the sak e of con v enience and to emphasize the adv ant ages of clique-width and NLC-width op erations, w e will u se clique-width exp ressions for the prob lems ind ep endent set and partition in to indep enden t sets and NLC-width expressions for the pr oblems clique and partition in to cliques. 15 4.2.1 Indep endence n umber Let us first consider the p r oblem of computing the indep endence n um b er (Problem 2.7) on graphs of b ou n ded clique-width. Let G be a graph defined b y some clique-width k -expression X . Let F ( X ) b e the (2 k − 1)- tuple ( . . . , a L , . . . ), wh ic h con tains for ev ery L ⊆ [ k ], L 6 = ∅ , a non negativ e intege r a L , whic h denotes the size of a largest indep endent set U in graph v al( X ) such that { lab( u ) | u ∈ U } = L . Then F ( X ) is b ounded in k ind ep endentl y of the size of X , b ecause F ( X ) has at exactly 2 k − 1 en tries. The follo wing observ ations sho w that F ( • i ) is compu table in time O (2 k ), F ( X ⊕ Y ) is compu table in time O (2 k ) from F ( X ) and F ( Y ), and F ( η i,j ( X )) and F ( ρ i → j ( X )) are computable in time O (2 k ) from F ( X ). 1. W e defin e F ( • i ) := ( . . . , a L , . . . ), wh ere ∀ L ⊆ [ k ] a L :=    1 if L = { i } 0 if L 6 = { i } . 2. Let F ( X ) = ( . . . , a L , . . . ) and F ( Y ) = ( . . . , b L , . . . ), then w e define F ( X ⊕ Y ) := ( . . . , c L , . . . ), wh ere c L := max L = L 1 ∪ L 2 a L 1 + b L 2 , L 1 , L 2 ⊆ [ k ]. 3. Let F ( X ) = ( . . . , a L , . . . ), then we define F ( η i,j ( X )) := ( . . . , b L , . . . ), wh ere ∀ L ⊆ [ k ] b L :=    a L if { i, j } * L 0 if { i, j } ⊆ L 4. Let F ( X ) = ( . . . , a L , . . . ), then we define F ( ρ i → j ( X )) := ( . . . , b L , . . . ), where ∀ L ⊆ [ k ] b L :=          a L if i 6∈ L and j 6∈ L max { a L , a L −{ j }∪{ i } } if i 6∈ L and j ∈ L 0 if i ∈ L Ob viously in graph v al( ρ i → j ( X )) there exists n o v ertex lab eled by i , thus for ev ery set L with i ∈ L w e k n o w that b L = 0 . After a d y n amic programming computation of F ( X ) we can compu te the size of a maxi- m um in dep end en t set in graph v al( X ) by α (v al( X )) := max a ∈ F ( X ) a . Theorem 4.3 The i ndep endenc e numb er of a gr aph of b ounde d clique-width c an b e c ompute d in line ar time, if the g r aph is given by som e clique-width k -expr ession. 4.2.2 Clique n um b er Let us next consider th e problem of computing the clique num b er (Problem 2.8). 16 In order ha ve some kno wledge on the order of the op erations 3 in a giv en expr ession X , w e assume that G is giv en b y some some NLC -width k -expression X . Let F ( X ) b e the (2 k − 1)- tuple ( . . . , a L , . . . ), wh ic h con tains for ev ery L ⊆ [ k ], L 6 = ∅ , a non negativ e intege r a L , whic h denotes the size of a largest clique C in graph v al( X ) suc h that { lab ( v ) | v ∈ C } = L . Then F ( X ) is b ounded in k indep endentl y from the size of X , b ecause F ( X ) has exactly 2 k − 1 en tries. Next we w ill sho w that that F ( • i ) is computable in time O (1), F ( X × S Y ) is computable in O (2 k ) time from F ( X ) and F ( Y ), and F ( ◦ R ( X )) is computable in time O (2 k ) from F ( X ). 1. W e defin e F ( • i ) := ( . . . , a L , . . . ), wh ere ∀ L ⊆ [ k ] a L :=    1 if L = { i } 0 if L 6 = { i } . 2. Let F ( X ) = ( . . . , a L , . . . ) and F ( Y ) = ( . . . , b L , . . . ), then we define F ( X × S Y ) := ( . . . , c L , . . . ), wh ere the v alues for c L are defined as f ollo ws. W e say rela tion S = { ( i 1 , j 1 ) , . . . , ( i l , j l ) } defines a join for the lab el sets L 1 , L 2 , denote d by L = L 1 ⊎ L 2 , if there is a subset I ⊆ [ l ] suc h that L 1 = ∪ i ′ ∈ I i i ′ and L 2 = ∪ i ′ ∈ I j i ′ and L = L 1 ∪ L 2 . Then w e can defin e ∀ L ⊆ [ k ] c L :=    max L = L 1 ⊎ L 2 { a L 1 + b L 2 , a L , b L } if there exists some L 1 ⊎ L 2 = L max { a L , b L } else 3. Let F ( X ) = ( . . . , a L , . . . ), then we define F ( ◦ R ( X )) := ( . . . , b L , . . . ), where ∀ L ⊆ [ k ] b L :=    max R ( L ′ )= L a L ′ if there exists s ome L ′ ⊆ [ k ] : R ( L ′ ) = L 0 else After a d y n amic programming computation of F ( X ) we can compu te the size of a maxi- m um clique in graph v al( X ) by ω (v al( X )) := max a ∈ F ( X ) a . Theorem 4.4 The clique numb er of a gr aph of b ounde d clique-width c an b e c ompute d in line ar time, if the g r aph is given by some k - expr ession. 4.2.3 Chromatic n um b er Next w e consider the pr ob lem of computing the c hromatic n um b er (Problem 2.9) on graphs of b ounded clique-width . Let G b e a graph give n b y some clique-width k -expression X . F or a disjoint partition of V G in to indep enden t sets V 1 , . . . , V r let M b e the m ulti set 4 h lab( V 1 ) , . . . , lab ( V r ) i . Let F ( X ) 3 In order to hand le clique- width expressions we w ant t o men tion the normal form for clique-width expres- sions defined in [EGW03]. 4 A multi set is a set that may ha ve severa l equal elements. F or a multi set with elements x 1 , . . . , x n w e write M = h x 1 , . . . , x n i . There is no ord er on the elements of M . The num b er ho w often an elemen t x occu rs in M is denoted by ψ ( M , x ). Two multi sets M 1 and M 2 are e qual if for each element x ∈ M 1 ∪ M 2 , ψ ( M 1 , x ) = ψ ( M 2 , x ), otherwise they are call ed differ ent . The empty multi set is denoted b y hi . The size of a m ulti set M is the n umber of its elements , denoted b y |M| . 17 b e the set of all m utually different multi sets M f or all disjoin t partitions of v ertex set V G in to ind ep endent sets. Then F ( X ) is p olynomially b ounded in the size of X , b ecause F ( X ) has at most ( | V G | + 1) 2 k − 1 m utually differen t multi sets eac h with at most | V G | nonemp ty subsets of [ k ]. The follo wing observ ations sh o w that F ( • i ) is computable in time O (1), F ( X ⊕ Y ) is computable in p olynomial time from F ( X ) and F ( Y ), and F ( η i,j ( X )) and F ( ρ i → j ( X )) are computable in p olynomial time from F ( X ). 1. W e defin e F ( • i ) := {h{ i }i } 2. Starting with set D := {hi} × F ( X ) × F ( Y ) extend D b y all triples t h at ca n b e obtained from some triple ( M , M ′ , M ′′ ) ∈ D b y remo ving a set L ′ from M ′ or a set L ′′ from M ′′ and inserting it into M , or by r emo ving b oth sets and ins erting L ′ ∪ L ′′ in to M . W e define F ( X ⊕ Y ) := {M | ( M , hi , hi ) ∈ D } . D gets at most ( | V G | + 1) 3(2 k − 1) triples and thus is compu table in p olynomial time. 3. W e defin e F ( η i,j ( X )) := {h L 1 , . . . , L r i ∈ F ( X ) | { i, j } 6⊆ L t for t = 1 , . . . , r } . 4. W e defin e F ( ρ i → j ( X )) := {h ρ i → j ( L 1 ) , . . . , ρ i → j ( L r ) i | h L 1 , . . . , L r i ∈ F ( X ) } . F or a relab eling ρ i → j let R i → j : [ k ] → [ k ] b e defi n ed by R i → j ( t ) := t if t 6 = i , and R i → j ( t ) := j if t = i . F or L ⊆ [ k ] let ρ i → j ( L ) := { R i → j ( t ) | t ∈ L } . There is a partition of the vertex set of v al( X ) in to r in dep end en t sets if and only if there is some M ∈ F ( X ) consisting of r lab el sets. The c hromatic num b er of graph v al( X ) can b e obtained by χ (v al( X )) : = min M∈ F ( X ) |M| . Theorem 4.5 The chr omatic numb er of a gr aph of b ounde d clique- width c an b e c ompute d i n p olynomial time. The time complexit y of compu ting the graph parameter c hromatic index (min imum n um- b er of c olors needed to colo r th e edges of a giv en graph) is op en up to no w ev en for co-g raph s. 4.2.4 Clique co v ering n umber Finally w e consider the problem of computin g the clique co v ering n umb er (Prob lem 2.10) on graphs of b ou n ded clique-width.. Let G b e a graph giv en b y some NLC-width k -expression X . F or a disjoin t partition of V G in to cliques V 1 , . . . , V r let M b e the m ulti set h lab( V 1 ) , . . . , lab( V r ) i . Let F ( X ) b e the set of all mutually differen t multi s ets M for all disjoin t partitions of ve rtex set V G in to cliques. Then F ( X ) is p olynomially b ounded in the size of X , b ecause F ( X ) has at most ( | V G | + 1) 2 k − 1 m utually differen t multi sets eac h with at most | V G | nonemp ty subsets of [ k ]. The follo wing observ ations sh o w that F ( • i ) is compu table in time O (1), F ( X × S Y ) is co mp utable in p olynomial time fr om F ( X ) and F ( Y ), and F ( ◦ R ( X )) is computable in p olynomial time from F ( X ). 1. W e defin e F ( • i ) := {h{ i }i } 18 2. In order to compu te F ( X × S Y ) from F ( X ) and F ( Y ) w e start with set D := {hi} × F ( X ) × F ( Y ) and extend D by all triples that can b e obtained fr om some triple ( M , M ′ , M ′′ ) ∈ D b y r emoving a set L ′ from M ′ or a s et L ′′ from M ′′ and in sert- ing it int o M , or if S defines a join for the lab el sets L ′ , L ′′ (defined in Section 4.2.2) b y remo ving b oth sets and in s erting L ′ ∪ L ′′ in to M . W e define F ( X × S Y ) := {M | ( M , hi , hi ) ∈ D } . D gets at most ( | V G | + 1) 3(2 k − 1) triples and thus is compu table in p olynomial time. 3. W e defin e F ( ◦ R ) := {h◦ R ( L 1 ) , . . . , ◦ R ( L r ) i | h L 1 , . . . , L r i ∈ F ( X ) } . F or a relab eling ◦ R and L ⊆ [ k ] let ◦ R ( L ) := { R ( t ) | t ∈ L } . There is a partitio n of the vertex set of v al( X ) in to r cliques if and only if there is some M ∈ F ( X ) consisting of r lab el sets. The clique co v ering num b er of graph v al( X ) can b e obtained by θ (v al( X )) := min M∈ F ( X ) |M| . Theorem 4.6 The clique c overing numb er of a gr aph of b ounde d clique-width c an b e c ompute d in p olynom ial time. 4.3 Computing α , ω , χ , and θ on co-graphs W e next sh o w that for graph s of clique-width at most 2 and graph s of NLC-width 1, i.e. for co-graphs (complemen t r educible graphs), our sho wn algorithms c an b e simplifi ed . A c o-g r aph is either • a sin gle v ertex (denoted b y • ), • the d isjoin t u nion of t wo co-g raph s G 1 , G 2 (denoted by G 1 ∪ G 2 ), or • the join of tw o co-graphs G 1 , G 2 , which connects ev ery v ertex of G 1 with ev ery vertex of G 2 (denoted by G 1 × G 2 ). Ob viously for ev ery co-graph w e can define a tree structure, denoted as c o-tr e e in [C PS85]. The lea ves of the co-tree rep r esen t the v ertices of the graph and the inner no d es of the co- tree corresp ond to the op erations applied on the sub expressions defi ned b y the t w o su btrees. Giv en s ome co-graph G we can construct a corresp onding co-tree T G in linear time by th e results sh o wn in [CPS85]. Using the tree structur e T G , based on the results of Corn eil et al. [CLSB81], we n ext giv e simple linear time algorithms for computing α , ω , χ , and θ on co-graphs. 4.3.1 Indep endence n umber F or ev ery co-graph G its indep end ence n umb er α ( G ) can recursively b e computed as follo ws. 1. If | V G | = 1, then α ( G ) := 1 . 2. If G = G 1 ∪ G 2 , then α ( G ) := α ( G 1 ) + α ( G 2 ). 3. If G = G 1 × G 2 , then α ( G ) := max { α ( G 1 ) , α ( G 2 ) } . Theorem 4.7 F or every c o-gr aph its indep endenc e numb er c an b e c ompute d in line ar time. 19 4.3.2 Clique n um b er F or every co-graph G its clique num b er ω ( G ) can recursiv ely b e computed as follo ws. 1. If | V G | = 1, then ω ( G ) := 1. 2. If G = G 1 ∪ G 2 , then ω ( G ) := max { ω ( G 1 ) , ω ( G 2 ) } . 3. If G = G 1 × G 2 , then ω ( G ) := ω ( G 1 ) + ω ( G 2 ). Theorem 4.8 F or every c o-gr aph its cliq ue numb er c an b e c ompute d in line ar time. 4.3.3 Chromatic n um b er Since co-graphs are p erfect, we kn o w that f or ev ery ind uced subgraph H of some co- graph G it holds χ ( H ) = ω ( H ), and thus w e can compute its c hromatic num b er χ ( G ) b y the same algorithm as shown for ω ( G ) ab ov e. Theorem 4.9 F or every c o-gr aph its chr omatic nu mb er c an b e c ompute d in line ar time. 4.3.4 Clique co v ering n umber Again, since co-graphs are p erfect, we know that for eve ry induced subgraph H of some co- graph G it holds θ ( H ) = α ( H ), and thus w e can compute clique co v ering num b er θ ( G ) by the same algorithm as sh o wn for α ( G ) ab ov e. Theorem 4.10 F or every c o-gr aph i ts clique c overing numb er c an b e c ompute d in line ar time. 4.4 Complemen t problems If L ⊆ CW k or L ⊆ NLC k , th en for the corresp onding set of complement graphs it holds L ⊆ CW 2 k [CO00] or L ⊆ NLC k [W an94], resp ectiv ely . This implies that f or eve ry graph problem Π solv able in p olynomial time on clique-width b ounded graphs, the corresp onding complemen t problem Π is also solv able in p olynomial time on clique-width b ounded graphs. F or example, in order to solv e the clique pr oblem on clique-width b ounded graph s one can use the data stru cture giv en in S ection 4.2.2. Alternativ ely , form a theoreticall y p oint of view, one could transf orm a giv en clique-width k -expression X for some give n graph G in to a clique-width 2 k -expression X ′ for its complemen t graph G , and app ly the algorithm for the indep endent set problem sh o wn in Section 4.2.1 on X ′ in order to obtain the v alue of α ( G ) = ω ( G ). The same holds true for the s olution of the partition in to cliques problem on clique-width b ou n ded graphs giv en in Section 4.2 .4 . W e can app ly the algorithm for the partition int o indep endent sets problem in Section 4.2.3 on an expr ession for the complement graph in order to obtain the v alue of χ ( G ) = θ ( G ). F rom a pr actica l point of view, one should prefer the solutions using a k -expr ession instead of those using a 2 k -expression, since the clique-width of the in put graph o ccurs as an exp onen t in the ru nning time of our fpt algorithms. 20 4.5 Clique-width and monadic second order logic On graph classes of b ounded clique-width, all graph p rop erties and optimization problems whic h are expressib le in monadic second order logic with quantificati ons o ver ve r tices and v ertex sets (MSO 1 -logic) are decidable in linear time if a clique-width expression for the graph is give n as an inpu t [CMR00]. T his also implies the existence of linear time algorithms for computing the indep endence num b er and clique num b er on graphs of b ound ed clique- width if a clique-width expression for th e graph is giv en as an in put. Note that the pr ob lems partition int o indep en den t sets and p artition in to cliques are not expressible in MSO 1 -logic. 5 Conclusions Let us b riefly discuss tw o fur ther w ell kno wn graph p arameters which can b e computed in p olynomial time on grap h s of b oun ded tree-width and graph s of b oun d ed clique-width. A do minating set for some graph G is a subset S ⊆ V G , suc h that e very vertex of V G − S is adjacen t to at least one vertex from S . The minimum v alue s such th at G has a d ominating set S ⊆ V G of size s is d enoted as the dominating numb er of graph G , denoted by γ ( G ). In [AP89] it is sho wn th at the dominating num b er of a graph of b ounded tree-width can b e computed in linear time. F ur ther in [Rao07] it is shown th at the dominating n u m b er of a graph of b oun ded clique-width can b e computed in p olynomial time. A vertex c over for some graph G is a subset S ⊆ V G , such that ev ery edge of G has at least one endp oint in S . The minimum v alue s such that G h as a v ertex co ve r S ⊆ V G of size s is denoted as the vertex c over numb er of graph G , d enoted b y τ ( G ). Gallai has sho wn in [Gal59] the follo wing relation b et w een the size of a minimal v ertex co v er and maximum indep end ent set of some graph G τ ( G ) + α ( G ) = | V G | , (1) whic h implies b y Theorem 3.3 that the v ertex co v er num b er of a graph of b ounded tree-width can b e computed in linear time. F or the same rea son b y Th eorem 4.3 the v ertex co ver n um b er of a graph of b oun ded clique-width can b e compu ted in linear time, if the graph is given by some clique-width k -expression. In this pap er we h a v e compared and illustrated ho w to u se th e tree structure of graphs of b oun d ed tree-width and graph s of b ounded clique-width to giv e t wo general dyn amic programming sc hemes to solv e problems along a tree decomp osition and along a clique-width expression. Let us fin ally emphasize that b oth approac h es are useful. On the one hand, clique-width allo ws to defin e larger classes of graphs of b ounded width than tree-width. On the other hand, there are graph problems which remain NP-complete on graph s of b ound ed clique-width, but which are fi x ed -parameter tractable on graphs of b ounded tree-width, such as the vertex disj oin t path problem whic h is d iscussed in [G W06 ]. F u rther the to ol of monadic second order logic allo w s to define p r o v able larger classes of p roblems whic h are solv able on tree-width b oun ded graph classes than on clique-width b oun ded graph classes, see [CMR00]. 21 References [A CP87] S. Arn b org, D.G. Corneil, and A. Prosku ro wski. Complexit y of fin ding em b ed- dings in a k -tree. SIAM Journal of Algebr aic and Discr ete Metho ds , 8(2):277– 284, 1987. [AP89] S. Arnborg and A. Prosku ro wski. Linear time algorithms f or NP-h ard problems restricted to partial k -trees. Discr ete Applie d M athematics , 23:11–2 4, 198 9. [Arn85] S. Arnb org. Efficien t algorithms for com b inatorial p roblems on graphs with b ound ed decomp osabilit y – A survey . BIT , 25:2–23 , 19 85. [BK07] H.L. Bo dlaender and A.M.C.A. K oster. Com binatorial optimization on graph s of b ound ed tree-width. Computer Journal , 2007. to app ear. [BM93] H.L. Bo dlaend er and R.H. M¨ ohring. The p ath width and tr eewidth of cographs. SIAM J. Disc. M ath. , 6(2):18 1–188, 1993 . [Bod 86] H.L. Bod laender. Classes of graphs with b oun ded treewidth . T echnical Rep ort R UU-CS-86-22, Universiteit Utrec ht, 1986. [Bod 87] H.L. Bod laender. Dynamic pr ogramming on graphs with b oun ded tree-width. T ec h nical Rep ort R UU-CS-87-22, Universiteit Utrec ht, 1987. [Bod 88a] H.L. Bo d laend er . Dynamic p r ogramming on graph s with b ounded tree-width. In Pr o c e e dings of International Col lo qu ium on A utomata, L anguages and Pr o gr am- ming , vo lume 317 of LNCS , pages 105–118. Sprin ger, 1988 . [Bod 88b] H.L. Bo dlaender. Planar graphs with b ou n ded treewidth. T ec hn ical Rep ort RUU- CS-88-14, Univ ersiteit Utrec ht , 19 88. [Bod 90] H.L. Bo dlaender. Pol yn omial algorithms for c hromatic index and graph isomor- phism on p artial k -trees. Journal of Algorithms , 11(4):6 31–643, 19 90. [Bod 96] H.L. Bo dlaender. A lin ear-time alg orithm for findin g tree-decomp ositions of sm all treewidth. SIAM Journal on Computing , 25(6):13 05–1317, 1996 . [Bod 98] H.L. Bod laender. A partial k -arb oretum of graphs with b ounded treewidth. The- or etic al Comp uter Scienc e , 209: 1–45, 1998. [CER91] B. Courcelle, J. Engelfriet, and G. Rozen b erg. Con text-free h andle-rewriting h yp ergraph grammars. In Gr aph-Gr ammars and Their A pplic ation to Computer Scienc e , v olume 532 of LNCS , pages 253–268. Springer, 1991. [CER93] B. Courcelle, J. En gelfriet, and G. Rozen b erg. Handle-rewriting hyp ergraph gram- mars. Journal of Computer and System Sci enc e s , 46:21 8–270, 1993. [CHL + 00] D.G. Corneil, M. Habib, J.M. Lanlignel, B. Reed, and U. Rotics. P olynomial time r ecognitio n of clique-width at most thr ee graph s. In P r o c e e dings of L atin Americ an Symp osium on The or etic al Informatics , v olume 1776 of LNCS , p ages 126–1 34. Springer, 200 0. 22 [Chl02] J. Chlebiko v a. Partial k -trees with maxim um c hr omatic num b er. D iscr e te Math- ematics , 259(1-3 ):269–276, 2002. [CLSB81] D.G. Corneil, H. Lerc h s, and L. Stewart-B u r lingham. Complement reducible graphs. Discr ete Applie d Mathema tics , 3:163–174 , 1981. [CM93] B. Courcelle and M. Mosbah. Monadic second-order ev aluations on tree- decomp osable graphs . The or etic al Comp uter Scienc e , 109:4 9–82, 199 3. [CMR00] B. Courcelle, J.A. Mak o wsky , an d U. Rotic s. Linear time solv ab le optimization problems on graph s of boun ded clique-width. The ory of Computing Systems , 33(2): 125–150, 20 00. [CO00] B. Courcelle and S. Olariu . Up p er b ounds to the clique width o f graphs. Discr e te Applie d Mathematics , 101:77 –114, 200 0. [CPS85] D.G. Corneil, Y. P erl, and L.K. Stew art. A linear recognition algorithm f or cographs. SIAM Journal on Computing , 14(4):9 26–934, 198 5. [CR05] D.G. Corneil and U. Rotics. On the relationship b et ween clique-width an d treewidth. SIAM Journal on Computing , 4:825–84 7, 200 5. [DF99] R.G. Do wney and M.R. F ello ws. Par ameterize d Complexity . Springer, New Y ork, 1999. [EGW01] W. Esp elage, F. Gurski, and E. W anke. How to solv e NP-hard graph problems on clique-width b ounded graphs in p olynomial time. In Pr o c e e dings of Gr aph - The or etic al Conc epts in Computer Scienc e , v olume 2 204 of LNCS , pag es 117–12 8. Springer, 2001. [EGW03] W. Esp elage, F. Gurski, and E. W ank e. Deciding clique-width for graphs of b ound ed tree-width. Journal of Gr aph Algorithms and Applic ations - Sp e cial Issue of J GA A on W A DS 2001 , 7(2):14 1–180, 2003 . [ER97] J. En gelfriet and G. Rozen b erg. No d e replacemen t graph grammars. In Hand- b o ok of Gr ammars and Computing by Gr aph T r ansformatio n , pages 1–94. W orld Scien tific, 1997. [F G06] J. Flum and M. Grohe. Par ameterize d Complexity The ory . Springer, Berlin, 2006 . [FRRS06] M.R. F ello ws, F.A. Rosam u nd, U. Rotics, and S. Szeider. Clique-width min i- mization is NP-hard. In Pr o c e e dings of the Annua l ACM Symp osium on The ory of Computing , p ages 354–3 62. ACM, 2006. [Gal59] T. Gallai. ¨ Ub er extreme Pu nkt- un d Kan tenmengen. Ann. U niv. Sci. Bu dap est, Eotvos Se ct. Math. , 2:133–13 8, 1959. [GJ79] M.R. Garey and D.S. Johnson. Computers and Intr actability: A Guide to the The ory of NP- Completeness . W.H. F reeman and Comp an y , San F rancisco, 1979. [GK03] M.U. Gerb er and D. Kobler. Algorithms for v ertex-partitioning problems on graphs w ith fi xed clique-width. The or etic al Com puter Scienc e , 299(1 -3):719–73 4 , 2003. 23 [GKS00] A. Gupta, D. Kaller, and T. Sh ermer. Linear-time algo rith m s for partial k -tree complemen ts. Algorithmic a , 27(3):2 54–274, 200 0. [GR00] M.C. Golum bic and U. Rotics. On th e cli qu e-width of some p erfect graph classes. International Journal of F oundations of Computer Scienc e , 11(3):42 3–443, 200 0. [Gup66] R.P . Gupta. Th e c hr omatic ind ex and the degree of a graph. Not. Amer. Math. So c. , 13:719, 1966. [Gur07] F. Gur ski. T he expressive p ow er and algorithmic use of graph paramet ers. Habili- tation T hesis, He inr ic h-Heine-Univ ersit¨ at D ¨ usseldorf, D ¨ usseldorf, Germany , 2007 . [GW00] F. Gurski and E. W ank e. Th e tree-width of clique-width b ounded graphs without K n,n . In Pr o c e e dings of Gr aph-The or etic al Conc epts in Computer Scienc e , vo lum e 1938 of LNCS , pages 196–205. Sprin ger, 2000. [GW05] F. Gurski and E . W anke. Minimizing NLC-width is NP-complete . In Pr o c e e dings of Gr aph-The or etic al Conc epts in Computer Scienc e , v olume 3787 of LNCS , pages 69–80 . Springer, 2005. [GW06] F. Gurski and E. W ank e. V ertex disjoin t paths on clique-width b ounded graph s. The or etic al Computer Scienc e , 359(1-3):18 8–199, 2006. [GW07] F. Gurski and E. W an ke. Line graphs of b ound ed clique-width. Discr e te Mathe- matics , 307(22):2 734–2754, 200 7. [Hag00 ] T . Hag eru p. Dynamic algo rithm s for graphs of b ounded treewidth. Algorithmic a , 27(3): 292–315, 20 00. [Hal76] R. Halin. S-fu nctions for graphs. J. Ge ometry , 8:171–1 76, 1976 . [HOSG07] P . Hlinen´ y, S. Oum, D. Seese, and G. Gottlob. Width parameters b ey ond tree- width and their ap p lications. Computer Journal , 2007. to app ear. [Hou06] S. Hougardy . Classes of p erfect graphs. Discr ete Mathematics , 306(19-2 0):2529– 2571, 2006. [INZ03] T. Ito, T. Nish izeki, and X. Zh ou. Algorithms f or multic olorings of partial k -trees. IEICE T r ans. on Information and Systems , E86-D:191–20 0, 2003. [Joh98] ¨ O. Johansson. Clique-decomp osition, NLC-d ecomp osition, and mo d ular decom- p osition - relationships and results for random g raph s. Congr essus Numer antium , 132:39 –60, 199 8. [KLM07] M. Kaminski, V.V. Lozin, and M. Milanic. Recent develo p m en ts on graph s of b ound ed cl iqu e-wid th . T echnical Rep ort RRR 06-20 07, Ru tgers Univ ersit y , 2007. [Klo94] T. Kloks. T r e e width, Computations and Appr oximations , volume 842 of LNCS . Springer, Berlin, 1994. [KR03] D. Kobler and U. Rotics. Edge dominating set and colorings on g r ap h s with fi xed clique-width. Discr ete Applie d M athematics , 126(2-3 ):197–221, 2003. 24 [KZN00] M.A. Kash em, X. Zh ou, a n d T. Nishizeki. Algorithms for generalized vertex- rankings of p artial k -trees. The or etic al Computer Scienc e , 240(2):40 7–427, 2000 . [LdMR07] V. Limouzy , F. de Mo ntgol fi er , and M. Rao. NLC-2 graph recognition and iso- morphism. In Pr o c e e dings of Gr aph - The or etic al Conc epts in Computer Scienc e , LNCS. Spr inger, 2007. to app ear. [MR99] J.A. Mak o wsky and U. Rotics. On the clique-width of graph s with few P 4 . Inter- national Journal of F oundations of Computer Scienc e , 10:329–3 48, 1999. [MRA G06] J.A. Mak o wsky , U. Rotics, I. Averb ouc h, and B. Go d li. C omputing graph p oly- nomials on graph s of b oun ded clique-width . In Pr o c e e dings of Gr aph-The or etic al Conc epts in Computer Scienc e , v olume 4271 of LNCS . Sp ringer, 2006 . [Nie06] R. Niedermeier. Invitation to Fixe d-Par ameter A lgorithms. Oxford Univ ersit y Press, New Y ork, 2006. [OS06] S. Oum and P .D. Seymour. Approxima ting clique-width and branch-width. J our- nal of Combinatorial The ory, Series B , 96(4):51 4–528, 200 6. [Oum05] S. Ou m. App ro ximating rank-wid th and clique-width quickly . In Pr o c e e dings of Gr aph-The or etic al Conc epts in Computer Scienc e , vo lum e 3787 of LNCS , p ages 49–58 . Springer, 2005. [Oum06] S. Oum. Appr oximati n g rank-width and clique-width quickly . Man uscrip t, 2006. [Rao06] M. Rao. D´ ecomp ositions de graphes et algorithmes efficaces. Phd th ` ese, Universit ´ e de Metz, 2006. [Rao07] M. Rao. MSOL p artitioning problems on graphs of b ounded treewidth a n d clique- width. The or etic al Computer Scienc e , 377(1-3) :260–267, 2007. [RS86] N. Rob ertson and P .D. S eymour. Grap h m in ors I I. Algo rithm ic asp ects of tree width. Journal of Algorithm s , 7:309–322 , 1986. [ST07] K . Suchan and I. T o din ca. On p o wers of graphs of b ounded NLC-width (clique- width). Discr ete Applie d Mathematics , 155(14) :1885–189 3 , 2007. [T o d 03] I. T o dinca. Coloring p o wers of graphs of b ounded clique-width. In Pr o c e e dings of Gr aph-The or etic al Conc epts in Computer Scienc e , vo lum e 2880 of LNCS , p ages 370–3 82. Springer, 200 3. [Viz64] V.G. Vizing. On an estimate of the c hromatic class of a p-graph. Meto dy D iskr et. Ana liz. , 3:917, 1964. [W an94] E. W ank e. k -NLC graphs and p olynomial algo r ith m s. Discr ete Applie d Mathe- matics , 54:251–2 66, 199 4. [W C83] J.A. W ald and C.J. Colb our n . Steiner trees, partial 2-trees, and minimum IFI net wo rks . Networks , 13:159–16 7, 1983. [ZFN00] X. Zhou, K . F use, and T. Nishizeki. A linear time algorithm for fin ding [ g , f ]- colorings of partial k -trees. A lgorithmic a , 27(3):227 –243, 2000 . 25