Partitioning Claw-Free Subcubic Graphs into Two Dominating Sets

A dominating set in a graph G is a set S subset of V(G) such that every vertex in V(G)\S has at least one neighbor in S. Let G be an arbitrary claw-free graph containing only vertices of degree two or three. In this paper, we prove that the vertex set of G can be partitioned into two dominating sets V-1 and V-2 such that for i = 1, 2, the subgraph of G induced by V-i is triangle-free and every vertex v is an element of V-i also has at least one neighbor in V-i if v has degree three in G. This gives an affirmative answer to a problem of Bacso et al. and generalizes a result of Desormeaux et al..