Grammars and clique-width bounds from split decompositions

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 graph grammar for distance-hereditary graphs consisting of four rules collected in a single equation. 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 decompositions (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.