Grammars and clique-width bounds from split decompositions

Abstract : Graph decompositions are important for algorithmic purposes and for graph structure theory. We relate the split decomposition introduced by Cunnigham to vertex substitution, graph grammars and clique-width. For this purpose, we extend the usual notion of substitution, upon which modular decomposition is based, by considering graphs with dead (or non-boundary) vertices. We obtain a simple grammar for distance-hereditary graphs. We also bound the clique-width of a graph in terms of those of the components of a split decomposition that need not be canonical. For extending these results to directed graphs and their split decompo-sitions (that we handle formally as graph-labelled trees), we need another extension of substitution : instead of two types of vertices, dead or alive as for undirected graphs, we need four types, in order to encode edge directions. We bound linearly the clique-width of a directed graph G in terms of the maximal clique-width of a component arising in a graph-labelled tree that defines G. This result concerns all directed graphs, not only the strongly connected ones considered by Cunningham.
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Submitted on : Wednesday, July 31, 2019 - 6:30:41 PM
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Bruno Courcelle. Grammars and clique-width bounds from split decompositions. Discrete Applied Mathematics, Elsevier, In press, ⟨10.1016/j.dam.2019.07.001⟩. ⟨hal-02093211v2⟩

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