Critically indecomposable graphs

• Authors: Chandan K. Dubey;Shashank K. Mehta
• Venue: Discrete Applied Mathematics
• Year: 2009

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.