Partitioning Claw-Free Subcubic Graphs into Two Dominating Sets

Abstract

A dominating set in a graph G is a set $S\subseteq V(G)$ such that every vertex in $V(G){\setminus } S$ 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 $v\in V_i$ 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 Bacsó et al. and generalizes a result of Desormeaux et al.