Split decomposition and graph-labelled trees: Characterizations and fully dynamic algorithms for totally decomposable graphs

In this paper, we revisit the split decomposition of graphs and give new combinatorial and algorithmic results for the class of totally decomposable graphs, also known as the distance hereditary graphs, and for two non-trivial subclasses, namely the cographs and the 3-leaf power graphs. Precisely, we give structural and incremental characterizations, leading to optimal fully dynamic recognition algorithms for vertex and edge modifications, for each of these classes. These results rely on the new combinatorial framework of graph-labelled trees used to represent the split decomposition of general graphs (and also the modular decomposition). The point of the paper is to use bijections between the aforementioned graph classes and graph-labelled trees whose nodes are labelled by cliques and stars. We mention that this bijective viewpoint yields directly an intersection model for the class of distance hereditary graphs.

► This paper propose new algorithmic and combinatorial results for graph classes which are totally decomposable for the split decomposition.
► We give strutural and (vertex/edge) incremental characterizations of distance hereditary graphs, cographs and 3-leaf power graphs
► We propose fully-dynamic recognition algorithms for distance hereditary graphs, cographs and 3-leaf power graphs
► These results rely on the new combinatorial framework of graph-labelled trees used to represent the split decomposition of general graphs.
► We use bijections between distance hereditary graphs and graph-labelled trees, yielding an intersection model for distance hereditary graphs.