Critically indecomposable graphs

Abstract

An undirected graph G = ( V , E ) with a specific subset X ⊂ V is called X -critical if G and G ( X ) , induced subgraph on X , are indecomposable but G ( V − { w } ) is decomposable for every w ∈ V − X . This is a generalization of critically indecomposable graphs studied by Schmerl and Trotter [J.H. Schmerl, W.T. Trotter, Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures, Discrete Mathematics 113 (1993) 191–205] and Bonizzoni [P. Bonizzoni, Primitive 2-structures with the ( n − 2 ) -property, Theoretical Computer Science 132 (1994) 151–178], who deal with the case where X is empty. We present several structural results for this class of graphs and show that in every X -critical graph the vertices of V − X can be partitioned into pairs ( a 1 , b 1 ) , ( a 2 , b 2 ) , … , ( a m , b m ) such that G ( V − { a j 1 , b j 1 , … , a j k , b j k } ) is also an X -critical graph for arbitrary set of indices { j 1 , … , j k } . These vertex pairs are called commutative elimination sequence. If G is an arbitrary indecomposable graph with an indecomposable induced subgraph G ( X ) , then the above result establishes the existence of an indecomposability preserving sequence of vertex pairs ( x 1 , y 1 ) , … , ( x t , y t ) such that x i , y i ∈ V − X . As an application of the commutative elimination sequence of an X -critical graph we present algorithms to extend a 3-coloring (similarly, 1-factor) of G ( X ) to entire G .