On indecomposability preserving elimination sequences

Abstract

A module of a graph is a non-empty subset of vertices such that every non-module vertex is either connected to all or none of the module vertices. An indecomposable graph contains no non-trivial module (modules of cardinality 1 and |V| are trivial). We present an algorithm to compute indecomposability preserving elimination sequence, which is faster by a factor of |V| compared to the algorithms based on earlier published work. The algorithm is based on a constructive proof of Ille's theorem [9]. The proof uses the properties of X-critical graphs, a generalization of critical indecomposable graphs. © Springer-Verlag Berlin Heidelberg 2006.