Discrete Mathematics 307 (2007) 756–759
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
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
© 2006 Elsevier B.V. All rights reserved.
Keywords: Clique-width; NLC-width; Line graphs; Co-gem-free line graphs
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 . 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 . The only essential differencebetween
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  by the existence of a
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 
and for graphs of bounded tree-width , 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)-free
E-mail addresses: firstname.lastname@example.org (F. Gurski), email@example.com (E. Wanke).
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.
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