Split Decomposition and Graph-Labelled Trees: Characterizations and Fully-Dynamic Algorithms for Totally Decomposable Graphs - LIRMM - Laboratoire d’Informatique, de Robotique et de Microélectronique de Montpellier Access content directly
Journal Articles Discrete Applied Mathematics Year : 2012

Split Decomposition and Graph-Labelled Trees: Characterizations and Fully-Dynamic Algorithms for Totally Decomposable Graphs

Emeric Gioan
Christophe Paul

Abstract

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 strutural and incremental characterizations, leading to optimal fullydynamic 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. 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.

Dates and versions

lirmm-00783420 , version 1 (01-02-2013)

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Emeric Gioan, Christophe Paul. Split Decomposition and Graph-Labelled Trees: Characterizations and Fully-Dynamic Algorithms for Totally Decomposable Graphs. Discrete Applied Mathematics, 2012, 160 (6), pp.708-733. ⟨10.1016/j.dam.2011.05.007⟩. ⟨lirmm-00783420⟩
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