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The band-, treeand lique-width are of primary importan e in algorithmi graph theory due to the fa t that many problems being NP-hard for general graphs an be solved in polynomial time when restri ted to graphs where one of these parameters is bounded. It is known that for any xed 3, all three parameters are unbounded for graphs with vertex degree at most . In this paper, we distinguish representative sub lasses of graphs with bounded vertex degree that have bounded band-, treeor lique-width. Our proofs are onstru tive and lead to eÆ ient algorithms for a variety of NP-hard graph problems when restri ted to those lasses. 1 Introdu tion The band-, treeand lique-width are of primary importan e in algorithmi graph theory due to the fa t that many problems whi h are NP-hard for general graphs an be solved in polynomial time when restri ted to graphs where one of these parameters is bounded. In fa t, all problems expressible in monadi se ond order logi be ome polynomial time solvable when restri ted to graphs of bounded tree-width [6, 14℄ and all problems expressible in monadi se ond order logi using quanti ers on verti es but not on edges be ome polynomial time solvable when restri ted to graphs of bounded lique-width [9, 10℄. This in ludes, for example, maximum lique, independent set, minimum dominating or independent dominating set problems as well as kolorability (for xed k), maximum indu ed mat hing, indu ed path, et . Furthermore, graphs of bounded band-width are easily seen to be of bounded treeand lique-width ( f. Proposition 1 below) and the mentioned problems are thus equally tra table. It is known that for any xed 3, all three parameters are unbounded for graphs with vertex degree at most . This is the ase, for instan e, for the soalled walls whi h are planar graphs of maximum vertex degree 3 ( f. [14℄) and have arbitrarily large band-, treeand lique-width [8, 14℄. The obje tive of the present paper is to distinguish sub lasses of graphs of bounded vertex degree that have bounded band-, treeor lique-width. To this end, we study hereditary lasses de ned by forbidding large indu ed subgraphs. Two parti ular forbidden graphs play the riti al role in our study. For integers i; j; k 0, these are the two graphs Si;j;k and Ti;j;k depi ted in Figure 1(a) and 1(b), respe tively. (Note that S0;j;k is a hordless path on j + k + 1 verti es and T0;0;0 is simply a triangle.) By S we shall denote the lass of graphs every omponent of whi h is of the form Si;j;k, and by T the lass of graphs every omponent of whi h is of the form Ti;j;k. rrrrrppp r r r r p p p r r r r p p p HH H Si;j;k (a) 12 i 1 i 1 2 j 1 j 1 2 k 1k r rrpppAA r r r r p p p r r r r p p p H H Ti;j;k (b) 01 i 1 i 0 1 j 1 j 0 1 k 1k Figure 1: The graphs (a) Si;j;k and (b) Ti;j;k We prove that in onne ted graphs of bounded vertex degree that do not ontain Si1;j1;k1 and Ti2;j2;k2 (for arbitrary values of the indi es) as indu ed subgraphs we an nd in polynomial time an indu ed path su h that all verti es are within bounded distan e of this path. This implies, in parti ular, that for any hereditary lass of graphs X with bounded vertex { 2 { degree that ontains neither S nor T , the tree-width and the lique-width are bounded by a onstant. Under the assumption that P 6= NP , this result is best possible for the lasses of graphs de ned by nitely many forbidden indu ed subgraphs. This is a onsequen e of the following two fa ts: rst, if a lass of graphs X de ned by nitely many forbidden indu ed subgraphs ontains S or T , then the independent dominating set problem is NP-hard in the lass X [3℄, and se ond, this problem is polynomially solvable for graphs with bounded tree-width or lique-width. For the band-width, we dis over some stronger onditions under whi h this parameter is bounded. All our results are algorithmi al in the sense that they not only prove boundedness but also imply the existen e of simple polynomial time algorithms for the respe tive widths problems. The outline of our presentation is as follows. In the next se tion we provide formal de nitions and prove preliminary results. In Se tion 3 we establish stru tural properties of graphs of bounded vertex degree that do not ontain Sk;k;k and Tk;k;k as indu ed subgraphs. Se tion 4 ontains our main results. In Se tion 5 we illustrate the results by applying them to several well-know lasses su h as asteroidal triple-free graphs (that in ludes omparability, permutation, and interval graphs) and hordal bipartite graphs. 2 De nitions and preliminary results For standard graph-theoreti al terminology, the reader is referred to any of the lassi al textbooks on graph theory. The set of verti es and set of edges of a graph G will be denoted by V (G) and E(G), respe tively. Given a vertex v 2 V (G), NG(v) stands for the neighborhood of v in the graph G, i.e. the set of verti es of G adja ent to v, and NG[v℄ for the losed neighborhood of v, i.e. NG[v℄ = NG(v)[ fvg. For a subset of verti es U V (G), G U is the subgraph of G obtained by deleting the verti es of U , NG(U) is the neighborhood of U , i.e. the set of verti es of G outside U that have a neighbor in U , and NG[U ℄ is the losed neighborhood of U , i.e. NG[U ℄ = NG(U) [ U . A graph G will be alled H-free if G does not ontain H as an indu ed subgraph. The lass of all graphs ontaining no indu ed subgraphs in a set M will be denoted Free(M). We shall write Freek(M) to denote the graphs with vertex degree at most k in Free(M). It is well-known that a lass of graphs X is hereditary (i.e. losed under deletion of verti es) if and only if X = Free(M) for a ertain set M (possibly in nite). For a lass of graphs Y, we denote by [Y℄k the lass of graphs G su h that G U belongs to Y for a subset U V (G) of ardinality at most k. Also, given a lass of graphs Y, [Y℄B denotes the lass of graphs every blo k of whi h belongs to Y. We now de ne the three width parameters that we onsider in the paper. Let L be an ordering of the verti es of a graph G, i.e. L is a bije tion from V (G) to f1; 2; : : : ; jV (G)jg. The width of the ordering is de ned as maxfjL(u) L(v)j j uv 2 E(G)g: The band-width bw(G) of G [11℄ is the minimum integer k for whi h there exists an ordering L of width k for G. { 3 { A tree de omposition [2, 14, 15℄ of a graph G is a pair (T;W) where T is a tree and W assigns a set Wt V (G) to ea h vertex t of T su h that (i) V (G) = St2V (T )Wt, (ii) for every edge uv 2 E(G), there is some t 2 V (T ) su h that u; v 2 Wt and (iii) for every vertex u 2 V (G), the set ft 2 V (T ) j u 2 Wtg indu es a subtree of T . The width of a tree de omposition (T;W) is maxt2V (T ) jWtj 1 and the tree-width tw(G) of G is the minimum width of a tree de omposition of G. Finally, the lique-width w(G) [7℄ of a graph G is the minimum number of labels needed to onstru t G using the following four operations: (i) Create a new vertex v with label i (denoted i(v)). (ii) Form the disjoint union of two labeled graphs G and H (denoted G H). (iii) Join all verti es with label i to all verti es with label j (i 6= j, denoted i;j). (iv) Change the label of all verti es with label i to j (denoted i!j). Every graph an be de ned by an algebrai expression using these four operations. For instan e, the y le on ve onse utive verti es a; b; ; d; e an be de ned as follows: 4;1( 4;3(4(e) 4!3( 3!2( 4;3(4(d) 3;2(3( ) 2;1(2(b) 1(a)))))))): Su h an expression is alled a k-expression if it uses at most k di erent labels. Thus the lique-width of G is the minimum k for whi h there exists a k-expression de ning G. After these de nitions we pro eed to some auxiliary results. Proposition 1 Let G be a graph and U V (G). Given an ordering of width k of the verti es of G U , one an onstru t in polynomial time (i) a tree de omposition of G of width at most 2dk2e+ jU j and (ii) a (k + 2 + jU j)-expression de ning G. Proof. Let n = jV (G) n U j. Let L be the ordering of width k of the verti es of G U and let vi 2 V (G) n U be su h that L(vi) = i for 1 i n. Let T be su h that V (T ) = V (G) n U and E(T ) = fvivi+1 j 1 i n 1g: For 1 i n, let Wvi = U [ (vj j 1 j n; jj ij &k2') : { 4 { Now it is easy to he k that (T;W) is a tree de omposition of G of width at most 2dk2e+ jU j whi h implies (i). To prove (ii) we des ribe a pro edure to onstru t G using the four operations des ribed above and k + 2 + jU j di erent labels. First, for all verti es u 2 U apply iu(u) su h that iu 6= iv for all u; v 2 U with u 6= v. Next, for all edges uv 2 E(G) with u; v 2 U apply iu;iv . Let I be a set of (k + 1) di erent labels su h that I \ fiu j u 2 Ug = ;: Let i0 be a label not in I [ fiu j u 2 Ug. For r from 1 up to n, apply the following sequen e of operations: Let is be the label presently assigned to vertex vs with maxf1; r kg s r. Let ir 2 I n fis j maxf1; r kg s rg: Apply ir(vr). For all u 2 NG(vr) \ U , apply ir;iu. For all vs 2 NG(vr) \ fv1; v2; : : : ; vr 1g, apply ir;is. Finally, apply ir k!i0 and pro eed to r + 1. It is obvious that this pro edure onstru ts G using k + 2 + jU j di erent labels whi h ompletes the proof. Corollary 1 If G is a graph and U V (G), then (i) tw(G) 2dbw(G U) 2 e+ jU j and (ii) w(G) bw(G U) + 2 + jU j. Proposition 2 If the tree-width of graphs in a lass Y is bounded by p, and a tree de omposition of width at most p for any graph in Y an be onstru ted in polynomial time, then one an onstru t in polynomial time a tree-de omposition of width at most p + k for any graph in [Y℄k and a tree-de omposition of width at most p for any graph in [Y℄B. Proof. For the graphs in [Y℄k, one an pro eed as in the proof of Proposition 1. Now let G 2 [Y℄B and let B1; B2; : : : ; Bl be the blo ks of G. Let (T (i);W(i)) be a tree de omposition of Bi for 1 i l. We assume that all graphs G; T (1); T (2); : : : ; T (l 1) and T (l) are vertex disjoint. Let C be the set of utverti es of G. For every utvertex u 2 C and every blo k Bi su h that u 2 V (Bi), we x an arbitrary vertex t(u; i) 2 V (T (i)) su h that u 2 W (i)t(u;i). Let T be the tree su h that V (T ) = C [ (Sli=1 V (T (i))), Sli=1E(T (i)) E(T ) and E(T ) ontains an edge ut(u; i) for every utvertex u 2 C and every blo k Bi su h that u 2 V (Bi). Let Wt = W (i)t for all t 2 V (T (i)), 1 i l and let Wu = fug for all u 2 B. It is straightforward to verify that (T;W) is a tree de omposition of width at most p of G and the proof is ompleted. Proposition 3 If the lique-width of graphs in a lass Y is bounded by p, and a p-expression for any graph in Y an be onstru ted in polynomial time, then one an onstru t in polynomial time a 2k(p + 1)-expression for any graph in [Y℄k and a (p + 2)-expression for any graph in [Y℄B. { 5 { Proof. To prove the result for graphs in [Y℄k, it suÆ es to onsider the ase k = 1. Let G 2 [Y℄1 and G fvg 2 Y. Given a p-expression F for G fvg, we rst onstru t a 2p-expression F 0 for G fvg in su h a way that every labeled vertex l(u) in F hanges its label either to l0(u) or to l00(u) in F 0 depending on whether or not it is adja ent to v. Now we only need one additional label to obtain a (2p + 1)-expression for G from F 0. Thus, w(G) 2 w(G fvg) + 1 and the proposition holds for graphs in [Y℄k by indu tion. To prove the result for graphs in [Y℄B, onsider a graph G 2 [Y℄B. By assumption, the lique-width of every blo k of G is bounded by p. We show by indu tion on the number of blo ks that w(G) p+2. Two additional labels needed to onstru t a (p+2)-expression for G will be denoted and . First, let G be a blo k and v be an arbitrary vertex in G. Any p-expression F (G) de ning G an trivially be modi ed into a (p + 1)-expression F 0 v(G) in whi h v is the only vertex labeled with . We then transform F 0 v(G) into a (p+2)-expression Fv(G) by re-labeling every vertex di erent from v with . Now let H be a graph with b > 1 blo ks and G be a blo k in H with a single utvertex v. By deleting from H all the verti es of G ex ept v, we obtain a graph with b 1 blo ks. Su h a graph an be de ned by a (p + 2)-expression T due to the indu tive hypothesis. Assume the vertex v is reated in T with label j. Then substituting j(v) with !j(Fv(G)) in T we obtain a (p + 2)-expression de ning H. To see this, it is enough to noti e that the label is never renamed or used in any -operation in T by the indu tive hypothesis. This ompletes the proof. 3 Properties of (Sk;k;k; Tk;k;k)-free graphs of bounded vertex degree In this se tion we establish a sequen e of useful stru tural and algorithmi al properties of graphs that do not ontain Sk;k;k and Tk;k;k as indu ed subgraphs and have bounded vertex degree. Throughout this se tion we let k and be xed positive integers. Lemma 1 If G is a Sk;k;k-free and Tk;k;k-free graph of vertex degree at most , then G does not ontain two vertex disjoint indu ed paths P : x0x1 : : : x4 k and Q : y0y1 : : : yk su h that y0x2 k 2 E(G) and all edges between verti es of P and verti es of Q are in ident to y0 (see Figure 2). Proof. For ontradi tion, we assume that two paths P and Q as in the statement exist. If x2 k is the only neighbor of y0 on P , then G is obviously not Sk;k;k-free, whi h is a ontradi tion. Hen e, for some 2 l 1, let NG(y0) \ V (P ) = fxi1 ; xi2 ; : : : ; xilg { 6 { q x0 q q : : : q q x2 kq : : : q q q x4 k q y0 q q ...q yk Q PFigure 2: P and Q su h that ij < ij+1 for ea h j = 1; : : : ; l 1. If ij+1 ij 2k for some 1 j l 1, then the subgraph of G indu ed by the set V (Q) [ fxij ; xij+1; : : : ; xij+(k 1)g [ fxij+1 ; xij+1 1; : : : ; xij+1 (k 1)g is isomorphi to Sk;k;k, whi h is a ontradi tion. Hen e ij+1 ij < 2k for ea h 1 j l 1. Sin e x2 k 2 NG(y0), this implies that i1 > 2 k ( 2)2k = 4k; il < 2 k + ( 2)2k = 4 k 4k and hen e the subgraph indu ed by the set V (Q) [ fxi1 ; xi1 1; : : : ; x0g [ fxil ; xil+1; : : : ; x4 kg ontains either Sk;k;k or Tk;k;k as an indu ed subgraph. This ontradi tion ompletes the proof. Lemma 2 Let G be a onne ted (Sk;k;k; Tk;k;k)-free graph of vertex degree at most , and P : x0x1 : : : x k an indu ed path in G. Then the subgraph G NG[V (P )℄ does not ontain an indu ed y le C : y1y2 : : : yly1 of length l 2( 1)(k + 1). Proof. For ontradi tion, we assume that G NG[V (P )℄ ontains an indu ed y le C as in the statement. Sin e G is onne ted, we may onsider a shortest path Q : z0z1 : : : zr onne ting P to C su h that z0 2 V (P ) and zr 2 V (C). Note that r 2. The vertex z1 has at most ( 1) neighbors in V (P ). These neighbors divide P into at most edge-disjoint segments, one of whi h has length at least k = k. Without loss of generality, we assume thatNG(z1) \ fxi; xi+1; : : : ; xi+k 1g = fxig { 7 { for some 0 i k k. If zr is the only neighbor of zr 1 in V (C), then G is obviously not Sk;k;k-free, whi h is a ontradi tion. Therefore, we assume that zr 1 has at least two neighbors in V (C). Clearly, zr 1 has at most ( 1) neighbors in V (C). These neighbors divide C into at least two and at most ( 1) edge-disjoint segments, one of whi h has length at least 2( 1)(k+1) 1 = 2k+2. Without loss of generality, we assume that NG(zr 1) \ fyj; yj+1; : : : ; yj+sg = fyj; yj+sg for some 1 j 2( 1)(k + 1) s and some s 2k + 2. Note that yjyj+s 2 E(G) is possible (s = l 1). Now the subgraph of G indu ed by the set (see Figure 3) fz1; : : : ; zr 1g [ fxi; xi+1; : : : ; xi+k 1g [ fyj; yj+1; : : : ; yj+kg [ fyj+s; yj+s 1; : : : ; yj+s kg ontains either Sk;k;k or Tk;k;k as an indu ed subgraph, whi h is a ontradi tion and the proof is ompleted. q yj q q : : : : : : : : : q q q yj+s Cqzr 1 ... Q q qq z1 : : : : : : q xi P : : : q q q xi+k 1 Figure 3: Parts of P , Q and C Corollary 2 If G is a onne ted (Sk;k;k; Tk;k;k)-free graph of vertex degree at most , then there is a set U V (G) of at most 1 = 1(k; ) verti es su h that G U ontains no indu ed y le of length at least 2( 1)(k + 1). Furthermore, su h a set an be found in polynomial time. Proof. If we de ne 01 = 01(k; ) = 1+P k 1 i=0 ( 1)i , then the graph G either has order jV (G)j 01 or ontains an indu ed path of length k. If jV (G)j 01, then let V 0 = V (G). If jV (G)j 01+1, then, by onsidering the pairwise distan es of the verti es of G, we an nd in polynomial time an indu ed path P of length k. Sin e G has maximum vertex degree at most , the set NG[V (P )℄ ontains at most 00 1 = 00 1(k; ) = ( k + 1)( 1) + 1 verti es. Let U = NG[V (P )℄. By Lemma 2, the graph G U ontains no indu ed y le of length at least 2( 1)(k + 1). The desired result now follows with 1 = maxf 01; 00 1g. { 8 { Lemma 3 If G is a onne ted (Sk;k;k; Tk;k;k)-free graph of vertex degree at most , then G ontains an indu ed path P su h that for every vertex u 2 V (G), distG(u; V (P )) = minfdistG(u; v) j v 2 V (P )g 4 k: Moreover, su h a path an be found in polynomial time. Proof. A pro edure that omputes a path with the desired property an be roughly des ribed as follows. (0) Find an arbitrary indu ed path P whi h is maximal under in lusion. (1) Denote by l the length of P , and by x0x1 : : : xl the verti es of P . For ea h vertex u 2 V (G), ompute distG(u; V (P )). If all these distan es are at most 4 k, then STOP: P is the path we sought for. (2) Let u 2 V (G) be a vertex of G with distG(u; V (P )) > 4 k. If l 4 k, then a shortest path P 0 from u to P is longer than P . In this ase, set P := P 0 and go to step (1). (3) By onsidering the pairwise distan e between u and the verti es of P , determine the set D = fv 2 V (P ) j distG(u; v) = distG(u; V (P ))g: Up to symmetry, we let r be the smallest index su h that xr 2 D and r l 2 k+1 (if there is no su h r, de ne r to be the largest index su h that xr 2 D and r 2 k 1). Let Q : u = y0y1 : : : ys = xr be a shortest path onne ting u to xr, where s = distG(u; V (P )) > 4 k. Set P := x2 kx2 k+1 : : : xrys 1ys 2 : : : y1u and go to step (1). Obviously, every step of the pro edure an be implemented in polynomial time. Now let us show that the loop (1)-(3) repeats at most jV (G)j times. First, noti e that sin e l > 4 k in step (3), D \ fx2 k; x2 k+1; : : : ; xl 2 kg = ;; otherwise G would ontain two vertex disjoint paths as des ribed in Lemma 1, whi h is a ontradi tion. Therefore, D fx0; x1; : : : ; x2 k 1g [ fxl; xl 1; : : : ; xl 2 k+1g: Clearly, D 6= ; and hen e in step (3) of the pro edure we nd a new indu ed path of length r 2 k + s > l 2 k + 1 2 k + 4 k + 1 > l: Summarizing, both in step (2) and step (3) we go to step (1) with a larger indu ed path. Therefore, after at most jV (G)j loops the pro edure terminates. This proves the orre tness of the pro edure and a polynomial time bound. { 9 { 4 Main Results We now pro eed to our main results. Theorem 1 If G is a Sk;k;k-free and Tk;k;k-free graph of vertex degree at most that does not ontain an indu ed y le of length at least 2( 1)(k + 1), then the band-width of G is bounded by a onstant 2 = 2(k; ), and an ordering of the verti es of G of width at most 2 an be determined in polynomial time. Proof. Clearly, we an deal with di erent omponents of G separately. Hen e we may assume that G is onne ted. By Lemma 3, we an nd in polynomial time an indu ed path P : x0x1 : : : xl su h that distG(u; V (P )) 4 k for every vertex u 2 V (G). We will now de ne a partition V (G) = l [ i=0Vi assigning every vertex of G to some vertex of the path P . For ea h i = 0; : : : ; l, the set Vi will onsist of the verti es assigned to xi 2 V (P ). q x0 V0 q x2 V2 q xi Vi q xl Vl : : : : : : Figure 4: The partition V (G) = V1 [ V2 [ : : : [ Vl Let u 2 V (G). As in the proof of Lemma 3, we determine in polynomial time the set D(u) = fv 2 V (P ) j distG(u; v) = distG(u; P )g: Sin e G has vertex degree at most and distG(u; V (P )) 4 k, we have jD(u)j 02 = 02(k; ) = 1 + 4 k X =0( 1) : We hoose an arbitrary vertex xi 2 D(u) and assign u to xi, i.e. u 2 Vi. The main property to establish the boundedness of the band-width is expressed in the following laim. Claim If u 2 Vi and v 2 Vj for some 0 i < j l su h that j i 2( 1)(k + 1) + 4( 1)(k + 1) 02; then uv 62 E(G). Proof of the laim: For ontradi tion, we assume that uv 2 E(G). { 10 { Let xi0 be the vertex in D(u) with the maximum index, then i0 i 2( 1)(k + 1) 02: To show this, onsider two onse utive verti es xs and xt in D(u), i.e. for ea h r 2 fs + 1; : : : ; t 1g, xr 62 D(u). Then t s < 2( 1)(k + 1), sin e otherwise the verti es of two shortest paths onne ting u to xs and xt together with the verti es xs; : : : ; xt would indu e a graph ontaining a hordless y le of length at least 2( 1)(k + 1). Analogously, if xj0 is the vertex in D(v) with the minimum index, then j j 0 2( 1)(k + 1) 02: Consequently, j 0 i0 j i 4( 1)(k + 1) 02 2( 1)(k + 1): But now the verti es of two shortest paths onne ting u and v respe tively to xi0 and xj0 together with the verti es xi0 ; xi0+1, : : : ; ; xj0 indu e in G a graph ontaining a hordless y le of length at least 2( 1)(k + 1). This ontradi tion ompletes the proof of the laim. Sin e G has vertex degree at most and by Lemma 3, we have jVij 02 = 02(k; ) for every 0 i l. Let v1v2 : : : vn be an ordering of the verti es of G in whi h u omes before v whenever u 2 Vi and v 2 Vj with i < j. By the above laim, it is obvious that vivj 2 E(G) implies jj ij 2 = (2( 1)(k + 1) + 4( 1)(k + 1) 02) 02 and the proof is ompleted. Whereas band-width is very sensitive to the deletion of a bounded number of verti es from the graph ( onsider e.g. the star K1;n 1), the tree-width and the lique-width are not. Combining Proposition 1, Corollary 2 and Theorem 1, we obtain the following orollary. Corollary 3 If G is a Sk;k;k-free and Tk;k;k-free graph of vertex degree at most , then (i) tw(G) 2d 2 2 e + 1 (ii) w(G) 2 + 2 + 1. Furthermore, a tree de omposition of G of width at most 2d 2 2 e + 1 and a ( 2 + 2 + 1)expression of G an be found in polynomial time. Note that given the on lusion (i) of Corollary 3, it follows from [1℄ or [13℄ that a tree de omposition of small width an be found eÆ iently. Nevertheless, the pro edure we proposed in the proof of Proposition 1 is mu h simpler. With the help of Propositions 2 and 3, a stronger result an be derived from Corollary 3 for the treeand lique-width of graphs with bounded vertex degree. Re all that S denotes the lass of graphs every omponent of whi h is of the form Si;j;k, and T the lass of graphs every omponent of whi h is of the form Ti;j;k. { 11 { Theorem 2 Let X be a hereditary lass of graphs with bounded vertex degree. If S 6 X and T 6 X , then (i) the tree-width of graphs in X is bounded by a onstant 3, and a tree de omposition of width at most 3 an be onstru ted in polynomial time and (ii) the lique-width of graphs in X is bounded by a onstant 4, and a 4-expression an be onstru ted in polynomial time. Proof. If S 6 X and T 6 X , we may onsider two graphs H 2 S n X and J 2 T n X su h that every omponent of H is of the form Sk;k;k and every omponent of J is of the form Tk;k;k for some xed k 1. Denote by s and t the number of omponents of H and J , respe tively. Also, let Hi and Ji denote, respe tively, the subgraphs of H and J indu ed by the verti es of exa tly i omponents. In parti ular, H1 = Sk;k;k, J1 = Tk;k;k, Hs = H and Jt = J . For every graph G with vertex degree at most , there are at most p = p(k; ) verti es in the losed neighborhood of set of verti es that indu es a subgraph isomorphi to either Sk;k;k or Tk;k;k. Therefore, the following in lusion holds. Free (Hs; Jt) Free (H1; J1) [ [Free (Hs 1; J1)℄p [[Free (H1; Jt 1)℄p [ [Free (Hs 1; Jt 1)℄2p: From this in lusion and Propositions 2 and 3 we on lude by indu tion on s and t that the tree-width and lique-width of graphs in the lass X Free (H; J) is bounded. The basis of indu tion is provided by Corollary 3. Noti e that s and t are onstants asso iated with the lass X . 5 Examples An asteroidal triple in a graph is an independent set of three verti es su h that for any two verti es in the set, there is a path between these two verti es avoiding the neighborhood of the third vertex [5℄. A graph is asteroidal triple-free, or AT-free for short, if it ontains no asteroidal triple. Asteroidal triple-free graphs in lude several well-known graph lasses su h as omparability graphs, permutation graphs, or interval graphs. All three graph parameters studied in this paper are unbounded in the lass of AT-free graphs (sin e they are unbounded even for permutation graphs [12℄), and several fundamental graph problems remain NPhard in that lass. However, a ording to our main results, the situation hanges ru ially whenever we deal with AT-free graphs with bounded vertex degree. Formally, Proposition 4 The band-, treeand lique-width of AT-free graphs of vertex degree at most is bounded by a onstant 5 = 5( ). { 12 { Proof. The proposition follows immediately from Theorem 1, Proposition 1 and the observation that S2;2;2, T2;2;2 and indu ed y les of length at least 6 ontain an asteroidal triple. As another important example, let us mention hordal bipartite graphs, i.e. bipartite graphs without indu ed y les of length at least 6. This lass has appli ations in the study of linear programming, as the bipartite adja en y matri es of hordal bipartite graphs are totally balan ed. Note that the lass of hordal bipartite graphs is not a sub lass of AT-free graphs, sin e S2;2;2 is not forbidden in this lass. The lique-width (and hen e the bandand tree-width) of hordal bipartite graphs in unbounded, sin e it is unbounded even for bipartite permutation graphs [4℄, a proper sub lass of hordal bipartite graphs. We use Propositions 2, 3 and Theorem 2 in order to show that hordal bipartite graphs with vertex degree at most three have bounded treeand liquewidth. Proposition 5 The treeand lique-width of hordal bipartite graphs with vertex degree at most three is bounded by a onstant. Proof. We shall prove the proposition by showing that every hordal bipartite graph G with vertex degree at most 3 ontaining S2;2;2 as an indu ed subgraph has a utvertex. We denote by x the enter of S2;2;2, by y1, y2 and y3 the verti es of degree 2, and by z1, z2 and z3 the respe tive verti es of degree 1 in S2;2;2. Assume y1 is not a utvertex in G, and onsider a shortest path P : p0p1 : : : pk onne ting the vertex z1 = p0 to a vertex pk 2 fy2; y3; z2; z3g in the subgraph G fy1g. If pk = z2, then k is even. Noti e that the even-indexed verti es of P di erent from pk are not adja ent to y2, sin e otherwise P is not a shortest path. Let j be the largest index with y1pj 2 E(G) (possibly j = 0). But now y1pj : : : pky2xy1 is an indu ed y le of length at least 6 in G, a ontradi tion. Analogously, pk 6= z3. If pk = y2, then k = 3. Indeed, be ause of the degree onstraint the y le C : p1 : : : pkxy1p1 ontains at most one hord, and this hord is of the form y1pj with an even j. Therefore, if k > 3, then the verti es of C indu e a hordless y le of length at least 6. Thus k = 3, and to avoid an indu ed y le C6, y1p2 2 E(G) and p1z2 62 E(G). But now y3 is a utvertex in G. Indeed, if not, there must exist a path in G fy3g onne ting z3 to a vertex in the set fz1; p1; z2g missing the verti es of the set B = fx; y1; y2; p2g (sin e all verti es in B have degree three in G), but then the verti es of the path together with some verti es in B and y3 would indu e a forbidden y le. Thus, the hordal bipartite graphs with vertex degree at most 3 form a sub lass of the lass [Free3(S2;2;2; T0;0;0)℄B. In the latter lass the treeand lique-width are bounded a ording to Propositions 2, 3 and Theorem 2. 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