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In this paper, we revisit the split decomposition of graphs and give newcombinatorial and algorithmic results for the class of totally decomposablegraphs, also known as the distance hereditary graphs, and for two non-trivialsubclasses, namely the cographs and the 3-leaf power graphs. Precisely, we givestrutural and incremental characterizations, leading to optimal fully-dynamicrecognition algorithms for vertex and edge modifications, for each of theseclasses. These results rely on a new framework to represent the splitdecomposition, namely the graph-labelled trees, which also captures the modulardecomposition of graphs and thereby unify these two decompositions techniques.The point of the paper is to use bijections between these graph classes andtrees whose nodes are labelled by cliques and stars. Doing so, we are also ableto derive an intersection model for distance hereditary graphs, which answersan open problem.

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