- Email: [email protected]
A local characterization of bounded clique-width for line graphs
Discrete Mathematics 307 (2007) 756 – 759 www.elsevier.com/locate/disc
Note
A local characterization of bounded clique-width for line graphs Frank Gurski, Egon Wanke Institute of Computer Science, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany Received 30 November 2005; received in revised form 3 July 2006; accepted 10 July 2006 Available online 21 August 2006
Abstract It is shown that a line graph G has clique-width at most 8k + 4 and NLC-width at most 4k + 3, if G contains a vertex whose non-neighbours induce a subgraph of clique-width k or NLC-width k in G, respectively. This relation implies that co-gem-free line graphs have clique-width at most 14 and NLC-width at most 7. It is also shown that in a line graph the neighbours of a vertex induce a subgraph of clique-width at most 4 and NLC-width at most 2. © 2006 Elsevier B.V. All rights reserved. Keywords: Clique-width; NLC-width; Line graphs; Co-gem-free line graphs
1. Introduction The clique-width of a graph G, denoted by clique-width(G), is the least integer k such that G can be defined by operations on vertex-labelled graphs using k labels [7]. These operations are the vertex disjoint union, the addition of edges between vertices controlled by a label pair, and the relabelling of vertices. The NLC-width of a graph G, denoted by NLC-width(G), is defined similarly in terms of closely related operations [13]. The only essential difference between the composition mechanisms of clique-width bounded graphs and NLC-width bounded graphs is the addition of edges. In an NLC-width composition the addition of edges is combined with the union operation. Both concepts are useful, because it is sometimes much more comfortable to use NLC-width expressions instead of clique-width expressions and vice versa, respectively. The concept of clique-width generalizes the well-known concept of tree-width defined in [12] by the existence of a tree-decomposition. Clique-width bounded graphs and tree-width bounded graphs are particularly interesting from an algorithmic point of view. Many NP-complete graph problems can be solved in polynomial time for graphs of bounded clique-width [5] and for graphs of bounded tree-width [6], respectively. There are many papers about the clique-width of graph classes defined by special forbidden graphs (Fig. 1), see e.g. [1–4]. One of the hardest proofs in these papers is that on the clique-width of (gem,co-gem)-free1 graphs.
E-mail addresses: [email protected] (F. Gurski), [email protected] (E. Wanke). 1 For a set of graphs F, a graph is said to be F-free if it does not contain a graph of F as an induced subgraph. 0012-365X/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2006.07.004
F. Gurski, E. Wanke / Discrete Mathematics 307 (2007) 756 – 759
757
Fig. 1. Special graphs.
Theorem 1 (Brandstädt et al. [3]). (Gem,co-gem)-free graphs have clique-width 16. Obviously, a graph G is (gem,co-gem)-free if and only if for every vertex of G the neighbours and non-neighbours induce a P4 -free graph, i.e. a co-graph, in G. Since co-graphs are exactly the graphs of clique-width at most 2 and NLC-width 1 [7,13], the question arises whether one can extend the result of Theorem 1 in the following way. Problem 2. Can the clique-width of a graph be bounded in terms of the clique-width of subgraphs induced by the neighbours and non-neighbours of certain vertices? In this paper we use the tight connection between the tree-width of a graph and the clique-width of its line graph shown in [8] to solve Problem 2 for line graphs. The line graph L(G) of a graph G has a vertex for every edge of G and an edge between two vertices if the corresponding edges of G are adjacent [14]. Graph G is called the root graph of L(G). We show that in a line graph the neighbours of a vertex induce a subgraph of clique-width at most 4 and NLC-width at most 2. Our main result is that a line graph G has clique-width at most 8k + 4 and NLC-width at most 4k + 3, if G contains a vertex whose non-neighbours induce a subgraph of clique-width k or NLC-width k in G, respectively. This result implies that certain classes of line graphs are of bounded clique-width. For example, co-gem-free line graphs have NLC-width at most 7 and clique-width at most 14. 2. Main results For a graph G = (VG , EG ) and a vertex u ∈ VG we define the set of neighbours of u in G by NG (u) = {v ∈ VG | {u, v} ∈ EG } and the set of non-neighbours of u in G by N G (u) = VG − ({u} ∪ NG (u)). Thus every vertex v of G defines a disjoint partition {v} ∪ NG (v) ∪ N G (v) of VG . Similarly we define for an edge e ∈ EG the set of neighbours of e in G by NG (e) = {e ∈ E | e ∩ e = {u} for some u ∈ VG } and the set of non-neighbours of e in G by N G (e) = EG − ({e} ∪ NG (e)). Again, every edge e of G defines a disjoint partition {e} ∪ NG (e) ∪ N G (e) of EG . Further, for a vertex set V ⊆ VG we define by G[V ] the subgraph of G induced by V and for an edge set E ⊆ EG we define by G[E ] the subgraph ({u ∈ VG | {u, v} ∈ E for some v ∈ VG }, E ) of G. Fig. 2 illustrates these notions. For a line graph L(G), by definition there exists a one-to-one mapping : VL(G) → EG between the vertices of L(G) and the edges of root graph G. It is easy to see from the definitions that for every vertex v ∈ VL(G) the following equations hold: L(G)[{v}] = L(G[(v)]), L(G)[NL(G) (v)] = L(G[NG ((v))]), and L(G)[N L(G) (v)] = L(G[N G ((v))]), see Fig. 2 for v = v1 .
Fig. 2. The figure shows a line graph L(G), its root graph G, and the corresponding bijection : VL(G) → EG : (vi ) = ei , 1 i 6. Further the sets NL(G) (v1 ) = {v2 , v3 } and NG (e1 ) = {e2 , e3 } are shown in light-grey and sets N L(G) (v1 ) = {v4 , v5 , v6 } and N G (e1 ) = {e4 , e5 , e6 } are shown in dark-grey.
758
F. Gurski, E. Wanke / Discrete Mathematics 307 (2007) 756 – 759
2.1. Clique-width of G[NG (v)] First we consider for a line graph G the clique-width of the subgraph induced by the neighbours of a vertex of G. Theorem 3. For every line graph G and every vertex v ∈ VG , NLC-width(G[NG (v)])2 and clique-width(G[NG (v)]) 4. Proof. Let G be a line graph, v ∈ VG , H be the corresponding root graph for G, and e = {u1 , u2 } ∈ EH , such that (v) = e. In order to define G[NG (v)] we consider the structure of H [NH (e)]. Graph H [NH (e)] contains at most vertices u1 and u2 , a number l of vertices only adjacent to u1 , a number m of vertices only adjacent to u2 , and a number n of vertices adjacent to u1 and u2 . Thus, the line graph of H [NH (e)], and by the observations above G[NG (v)], is an induced subgraph of two disjoint cliques Kn+m and Kn+l which are connected by n vertex disjoint edges. This graph can easily be constructed by NLC-width operations using one label for each of the two cliques. Since the clique-width of a graph is at most twice its NLC-width [10] the result follows. 2.2. Bounded clique-width of G[N G (v)] implies bounded clique-width for G We now will show that for every line graph G and every vertex v ∈ VG the clique-width of G can be bounded by the clique-width of graph G[N G (v)]. Lemma 4. Let G be a graph, e ∈ EG , such that graph G[N G (e)] has tree-width at most k. Then G has tree-width at most k + 2. Proof. Let G be a graph and e = {u1 , u2 } ∈ EG , such that G[N G (e)] has tree-width at most k. By the definition of tree-width [12] there is a tree-decomposition (X = {Xu ⊆ VG[N G (e)] | u ∈ VT }, T = (VT , ET )) for G[N G (e)], such that |Xu | k + 1. It is easy to verify that (X = {Xu ∪ {u1 , u2 } | Xu ∈ X}, T ) is a tree-decomposition for graph G, such that |Xu | k + 3, i.e. G has tree-width at most k + 2. The last lemma can be used to show our main result in a very simple way because it is based on the results of [8]. Theorem 5. Let G be a line graph, v ∈ VG , such that graph G[N G (v)] has NLC-width at most k (clique-width at most k). Then G has NLC-width at most 4k + 3 (clique-width at most 8k + 4, respectively). Proof. Let G be a line graph with root graph H, v ∈ VG , such that NLC-width(G[N G (v)]) k, and let e ∈ EH such that (v)=e. Since graph G[N G (v))] has NLC-width at most k, the corresponding root graph H [N H (e)] has tree-width at most 4k − 1 [8]. Thus by Lemma 4 graph H has tree-width at most 4k + 1 and graph G has NLC-width at most 4k + 3 [8]. A similar argumentation shows that clique-width(G[N G (v)]) k implies that clique-width(G)2 · (4k − 1 + 2) + 2 = 8k + 4. Since graphs of bounded clique-width are closed under taking induced subgraphs we can conclude the following characterization. Corollary 6. A set of line graphs L has bounded clique-width if and only if for every graph G ∈ L there exists a vertex vG such that {G[N G (vG )] | G ∈ L} has bounded clique-width. Since P4 -free graphs have NLC-width 1 [13], Theorem 5 and the relation between clique-width and NLC-width [10] imply the following bounds. Corollary 7. Co-gem-free line graphs have NLC-width at most 7 and clique-width at most 14.
F. Gurski, E. Wanke / Discrete Mathematics 307 (2007) 756 – 759
759
3. Conclusions Since Lemma 4 also holds for path-width2 instead of tree-width, the root graph of a line graph of linear cliquewidth3 at most k has path-width at most 4k − 1 [8], and the line graph of a graph of path-width k has linear clique-width at most 2k + 1 [8], the proof of Theorem 5 also shows that for every line graph G and every vertex v ∈ VG the linear clique-width of G can be bounded by the linear clique-width of graph G[N G (v)]. It remains open whether for arbitrary graphs the clique-width can be bounded by considering the clique-width of the graphs induced by the neighbours and non-neighbours of certain vertices. In any case Theorems 3 and 5 do not hold for arbitrary graphs. Acknowledgements The authors want to thank the anonymous referees for their helpful remarks. References [1] A. Brandstädt, F.F. Dragan, H.-O. Le, R. Mosca, New graph classes of bounded clique width, in: Proceedings of Graph–Theoretical Concepts in Computer Science, Lecture Notes in Computer Sciences, vol. 2573, Springer, Berlin, 2002, pp. 57–67. [2] A. Brandstädt, J. Engelfriet, H.-O. Le, V.V. Lozin, Clique-width for four-vertex forbidden subgraphs, in: Proceedings of the International Symposium on Fundamentals of Computation Theory, Lecture Notes in Computer Science, vol. 3623, Springer, Berlin, 2005. [3] A. Brandstädt, H.O. Le, R. Mosca, Gem- and co-gem-free graphs have bounded clique-width, Internat. J. Found. Comput. Sci. 15 (2004) 163–185. [4] A. Brandstädt, H.O. Le, R. Mosca, Chordal co-gem-free and (P5 ,gem)-free graphs have bounded clique-width, Discrete Appl. Math. 145 (2005) 232–241. [5] B. Courcelle, J.A. Makowsky, U. Rotics, Linear time solvable optimization problems on graphs of bounded clique-width, Theory Comput. Systems 33 (2) (2000) 125–150. [6] B. Courcelle, M. Mosbah, Monadic second-order evaluations on tree-decomposable graphs, Theoret. Comput. Sci. 109 (1993) 49–82. [7] B. Courcelle, S. Olariu, Upper bounds to the clique width of graphs, Discrete Appl. Math. 101 (2000) 77–114. [8] F. Gurski, E. Wanke, Minimizing NLC-width is NP-complete (Extended Abstract), in: Proceedings of Graph–Theoretical Concepts in Computer Science, Lecture Notes in Computer Science, vol. 3787, Springer, Berlin, 2005, pp. 69–80. [9] F. Gurski, E. Wanke, On the relationship between NLC-width and linear NLC-width, Theoret. Comput. Sci. 347 (1–2) (2005) 76–89. [10] Ö. Johansson, Clique-decomposition, NLC-decomposition, and modular decomposition—relationships and results for random graphs, Congr. Numer. 132 (1998) 39–60. [11] N. Robertson, P.D. Seymour, Graph minors I, Excluding a forest, J. Combin. Theory Ser. B 35 (1983) 39–61. [12] N. Robertson, P.D. Seymour, Graph minors II, Algorithmic aspects of tree width, J. Algorithms 7 (1986) 309–322. [13] E. Wanke, k-NLC graphs and polynomial algorithms, Discrete Appl. Math. 54 (1994) 251–266. [14] H. Whitney, Congruent graphs and the connectivity of graphs, Amer. J. Math. 54 (1932) 150–168.
2 Path-width was defined in [11] similar to tree-width, the only difference is that the underlying graph of the tree-decomposition is a path instead of a tree. 3 Linear clique-width was defined in [9] similar to clique-width, the only difference is that at least one argument of every disjoint union operation is a single labelled vertex.
Note
A local characterization of bounded clique-width for line graphs Frank Gurski, Egon Wanke Institute of Computer Science, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany Received 30 November 2005; received in revised form 3 July 2006; accepted 10 July 2006 Available online 21 August 2006
Abstract It is shown that a line graph G has clique-width at most 8k + 4 and NLC-width at most 4k + 3, if G contains a vertex whose non-neighbours induce a subgraph of clique-width k or NLC-width k in G, respectively. This relation implies that co-gem-free line graphs have clique-width at most 14 and NLC-width at most 7. It is also shown that in a line graph the neighbours of a vertex induce a subgraph of clique-width at most 4 and NLC-width at most 2. © 2006 Elsevier B.V. All rights reserved. Keywords: Clique-width; NLC-width; Line graphs; Co-gem-free line graphs
1. Introduction The clique-width of a graph G, denoted by clique-width(G), is the least integer k such that G can be defined by operations on vertex-labelled graphs using k labels [7]. These operations are the vertex disjoint union, the addition of edges between vertices controlled by a label pair, and the relabelling of vertices. The NLC-width of a graph G, denoted by NLC-width(G), is defined similarly in terms of closely related operations [13]. The only essential difference between the composition mechanisms of clique-width bounded graphs and NLC-width bounded graphs is the addition of edges. In an NLC-width composition the addition of edges is combined with the union operation. Both concepts are useful, because it is sometimes much more comfortable to use NLC-width expressions instead of clique-width expressions and vice versa, respectively. The concept of clique-width generalizes the well-known concept of tree-width defined in [12] by the existence of a tree-decomposition. Clique-width bounded graphs and tree-width bounded graphs are particularly interesting from an algorithmic point of view. Many NP-complete graph problems can be solved in polynomial time for graphs of bounded clique-width [5] and for graphs of bounded tree-width [6], respectively. There are many papers about the clique-width of graph classes defined by special forbidden graphs (Fig. 1), see e.g. [1–4]. One of the hardest proofs in these papers is that on the clique-width of (gem,co-gem)-free1 graphs.
E-mail addresses: [email protected] (F. Gurski), [email protected] (E. Wanke). 1 For a set of graphs F, a graph is said to be F-free if it does not contain a graph of F as an induced subgraph. 0012-365X/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2006.07.004
F. Gurski, E. Wanke / Discrete Mathematics 307 (2007) 756 – 759
757
Fig. 1. Special graphs.
Theorem 1 (Brandstädt et al. [3]). (Gem,co-gem)-free graphs have clique-width 16. Obviously, a graph G is (gem,co-gem)-free if and only if for every vertex of G the neighbours and non-neighbours induce a P4 -free graph, i.e. a co-graph, in G. Since co-graphs are exactly the graphs of clique-width at most 2 and NLC-width 1 [7,13], the question arises whether one can extend the result of Theorem 1 in the following way. Problem 2. Can the clique-width of a graph be bounded in terms of the clique-width of subgraphs induced by the neighbours and non-neighbours of certain vertices? In this paper we use the tight connection between the tree-width of a graph and the clique-width of its line graph shown in [8] to solve Problem 2 for line graphs. The line graph L(G) of a graph G has a vertex for every edge of G and an edge between two vertices if the corresponding edges of G are adjacent [14]. Graph G is called the root graph of L(G). We show that in a line graph the neighbours of a vertex induce a subgraph of clique-width at most 4 and NLC-width at most 2. Our main result is that a line graph G has clique-width at most 8k + 4 and NLC-width at most 4k + 3, if G contains a vertex whose non-neighbours induce a subgraph of clique-width k or NLC-width k in G, respectively. This result implies that certain classes of line graphs are of bounded clique-width. For example, co-gem-free line graphs have NLC-width at most 7 and clique-width at most 14. 2. Main results For a graph G = (VG , EG ) and a vertex u ∈ VG we define the set of neighbours of u in G by NG (u) = {v ∈ VG | {u, v} ∈ EG } and the set of non-neighbours of u in G by N G (u) = VG − ({u} ∪ NG (u)). Thus every vertex v of G defines a disjoint partition {v} ∪ NG (v) ∪ N G (v) of VG . Similarly we define for an edge e ∈ EG the set of neighbours of e in G by NG (e) = {e ∈ E | e ∩ e = {u} for some u ∈ VG } and the set of non-neighbours of e in G by N G (e) = EG − ({e} ∪ NG (e)). Again, every edge e of G defines a disjoint partition {e} ∪ NG (e) ∪ N G (e) of EG . Further, for a vertex set V ⊆ VG we define by G[V ] the subgraph of G induced by V and for an edge set E ⊆ EG we define by G[E ] the subgraph ({u ∈ VG | {u, v} ∈ E for some v ∈ VG }, E ) of G. Fig. 2 illustrates these notions. For a line graph L(G), by definition there exists a one-to-one mapping : VL(G) → EG between the vertices of L(G) and the edges of root graph G. It is easy to see from the definitions that for every vertex v ∈ VL(G) the following equations hold: L(G)[{v}] = L(G[(v)]), L(G)[NL(G) (v)] = L(G[NG ((v))]), and L(G)[N L(G) (v)] = L(G[N G ((v))]), see Fig. 2 for v = v1 .
Fig. 2. The figure shows a line graph L(G), its root graph G, and the corresponding bijection : VL(G) → EG : (vi ) = ei , 1 i 6. Further the sets NL(G) (v1 ) = {v2 , v3 } and NG (e1 ) = {e2 , e3 } are shown in light-grey and sets N L(G) (v1 ) = {v4 , v5 , v6 } and N G (e1 ) = {e4 , e5 , e6 } are shown in dark-grey.
758
F. Gurski, E. Wanke / Discrete Mathematics 307 (2007) 756 – 759
2.1. Clique-width of G[NG (v)] First we consider for a line graph G the clique-width of the subgraph induced by the neighbours of a vertex of G. Theorem 3. For every line graph G and every vertex v ∈ VG , NLC-width(G[NG (v)])2 and clique-width(G[NG (v)]) 4. Proof. Let G be a line graph, v ∈ VG , H be the corresponding root graph for G, and e = {u1 , u2 } ∈ EH , such that (v) = e. In order to define G[NG (v)] we consider the structure of H [NH (e)]. Graph H [NH (e)] contains at most vertices u1 and u2 , a number l of vertices only adjacent to u1 , a number m of vertices only adjacent to u2 , and a number n of vertices adjacent to u1 and u2 . Thus, the line graph of H [NH (e)], and by the observations above G[NG (v)], is an induced subgraph of two disjoint cliques Kn+m and Kn+l which are connected by n vertex disjoint edges. This graph can easily be constructed by NLC-width operations using one label for each of the two cliques. Since the clique-width of a graph is at most twice its NLC-width [10] the result follows. 2.2. Bounded clique-width of G[N G (v)] implies bounded clique-width for G We now will show that for every line graph G and every vertex v ∈ VG the clique-width of G can be bounded by the clique-width of graph G[N G (v)]. Lemma 4. Let G be a graph, e ∈ EG , such that graph G[N G (e)] has tree-width at most k. Then G has tree-width at most k + 2. Proof. Let G be a graph and e = {u1 , u2 } ∈ EG , such that G[N G (e)] has tree-width at most k. By the definition of tree-width [12] there is a tree-decomposition (X = {Xu ⊆ VG[N G (e)] | u ∈ VT }, T = (VT , ET )) for G[N G (e)], such that |Xu | k + 1. It is easy to verify that (X = {Xu ∪ {u1 , u2 } | Xu ∈ X}, T ) is a tree-decomposition for graph G, such that |Xu | k + 3, i.e. G has tree-width at most k + 2. The last lemma can be used to show our main result in a very simple way because it is based on the results of [8]. Theorem 5. Let G be a line graph, v ∈ VG , such that graph G[N G (v)] has NLC-width at most k (clique-width at most k). Then G has NLC-width at most 4k + 3 (clique-width at most 8k + 4, respectively). Proof. Let G be a line graph with root graph H, v ∈ VG , such that NLC-width(G[N G (v)]) k, and let e ∈ EH such that (v)=e. Since graph G[N G (v))] has NLC-width at most k, the corresponding root graph H [N H (e)] has tree-width at most 4k − 1 [8]. Thus by Lemma 4 graph H has tree-width at most 4k + 1 and graph G has NLC-width at most 4k + 3 [8]. A similar argumentation shows that clique-width(G[N G (v)]) k implies that clique-width(G)2 · (4k − 1 + 2) + 2 = 8k + 4. Since graphs of bounded clique-width are closed under taking induced subgraphs we can conclude the following characterization. Corollary 6. A set of line graphs L has bounded clique-width if and only if for every graph G ∈ L there exists a vertex vG such that {G[N G (vG )] | G ∈ L} has bounded clique-width. Since P4 -free graphs have NLC-width 1 [13], Theorem 5 and the relation between clique-width and NLC-width [10] imply the following bounds. Corollary 7. Co-gem-free line graphs have NLC-width at most 7 and clique-width at most 14.
F. Gurski, E. Wanke / Discrete Mathematics 307 (2007) 756 – 759
759
3. Conclusions Since Lemma 4 also holds for path-width2 instead of tree-width, the root graph of a line graph of linear cliquewidth3 at most k has path-width at most 4k − 1 [8], and the line graph of a graph of path-width k has linear clique-width at most 2k + 1 [8], the proof of Theorem 5 also shows that for every line graph G and every vertex v ∈ VG the linear clique-width of G can be bounded by the linear clique-width of graph G[N G (v)]. It remains open whether for arbitrary graphs the clique-width can be bounded by considering the clique-width of the graphs induced by the neighbours and non-neighbours of certain vertices. In any case Theorems 3 and 5 do not hold for arbitrary graphs. Acknowledgements The authors want to thank the anonymous referees for their helpful remarks. References [1] A. Brandstädt, F.F. Dragan, H.-O. Le, R. Mosca, New graph classes of bounded clique width, in: Proceedings of Graph–Theoretical Concepts in Computer Science, Lecture Notes in Computer Sciences, vol. 2573, Springer, Berlin, 2002, pp. 57–67. [2] A. Brandstädt, J. Engelfriet, H.-O. Le, V.V. Lozin, Clique-width for four-vertex forbidden subgraphs, in: Proceedings of the International Symposium on Fundamentals of Computation Theory, Lecture Notes in Computer Science, vol. 3623, Springer, Berlin, 2005. [3] A. Brandstädt, H.O. Le, R. Mosca, Gem- and co-gem-free graphs have bounded clique-width, Internat. J. Found. Comput. Sci. 15 (2004) 163–185. [4] A. Brandstädt, H.O. Le, R. Mosca, Chordal co-gem-free and (P5 ,gem)-free graphs have bounded clique-width, Discrete Appl. Math. 145 (2005) 232–241. [5] B. Courcelle, J.A. Makowsky, U. Rotics, Linear time solvable optimization problems on graphs of bounded clique-width, Theory Comput. Systems 33 (2) (2000) 125–150. [6] B. Courcelle, M. Mosbah, Monadic second-order evaluations on tree-decomposable graphs, Theoret. Comput. Sci. 109 (1993) 49–82. [7] B. Courcelle, S. Olariu, Upper bounds to the clique width of graphs, Discrete Appl. Math. 101 (2000) 77–114. [8] F. Gurski, E. Wanke, Minimizing NLC-width is NP-complete (Extended Abstract), in: Proceedings of Graph–Theoretical Concepts in Computer Science, Lecture Notes in Computer Science, vol. 3787, Springer, Berlin, 2005, pp. 69–80. [9] F. Gurski, E. Wanke, On the relationship between NLC-width and linear NLC-width, Theoret. Comput. Sci. 347 (1–2) (2005) 76–89. [10] Ö. Johansson, Clique-decomposition, NLC-decomposition, and modular decomposition—relationships and results for random graphs, Congr. Numer. 132 (1998) 39–60. [11] N. Robertson, P.D. Seymour, Graph minors I, Excluding a forest, J. Combin. Theory Ser. B 35 (1983) 39–61. [12] N. Robertson, P.D. Seymour, Graph minors II, Algorithmic aspects of tree width, J. Algorithms 7 (1986) 309–322. [13] E. Wanke, k-NLC graphs and polynomial algorithms, Discrete Appl. Math. 54 (1994) 251–266. [14] H. Whitney, Congruent graphs and the connectivity of graphs, Amer. J. Math. 54 (1932) 150–168.
2 Path-width was defined in [11] similar to tree-width, the only difference is that the underlying graph of the tree-decomposition is a path instead of a tree. 3 Linear clique-width was defined in [9] similar to clique-width, the only difference is that at least one argument of every disjoint union operation is a single labelled vertex.