Living Reference Work Entry

Encyclopedia of Algorithms

pp 1-7

Date: Latest Version

Split Decomposition via Graph-Labelled Trees

  • Christophe PaulAffiliated withCNRS, Laboratoire d’Informatique Robotique et Microélectronique de Montpellier, Université Montpellier 2 Email author 

Keywords

Split decomposition LexBFS Circle graphs Distance hereditary graphs Permutation graphs Parity graphs

Problem Definition

Years and Authors of Summarized Original Work

2012; Gioan, Paul

2014; Gioan, Paul, Tedder, Corneil

Introduced by Cunningham and Edmonds [11], the split decomposition, also known as the join (or 1-join) decomposition, ranges among the classical graph decomposition schemes. Given a graph G = (V, E), a bipartition (A, B) of the vertex set V (with \(\vert A\vert \geqslant 2\) and \(\vert B\vert \geqslant 2\)) is a split if there are subsets A′A and B′B, called frontiers, such that there is an edge between a vertex uA and vB if and only if uA′ and vB′ (see Fig. 1). A graph is prime if it does not contain any split. Observe that an induced cycle of length at least 5 is a prime graph. A graph is degenerate if every bipartition (A, B) with | A | 2 and | B | 2 is a split. It can be shown that a degenerate graphs are either cliques or stars. The split decomposition consists in recursively decompose a graph into a set of disjoint graphs ...

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