A Local Characterization of Bounded Clique-Width for Line Graphs Frank Gurski, Egon Wanke

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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.

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 ,1i 6. Further the sets NL(G)(v1) ={v2,v3} and NG(e1) ={e2,e3} are shown in light-grey and sets NL(G)(v1) ={v4,v5,v6} and NG(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)],isan 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. LetGbeagraph, 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 (X ={X ⊆ V | u ∈ V },T = (V ,E )) G[N (e)] tree-width [12] there is a tree-decomposition u G[NG(e)] T T T for G , 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 clique- width3 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

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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.