Partitioning Claw-Free Subcubic Graphs into Two Dominating Sets

A dominating set in a graph G is a set $$S\subseteq V(G)$$ S ⊆ V ( G ) such that every vertex in $$V(G){\setminus } S$$ 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 $$V_1$$ V 1 and $$V_2$$ V 2 such that for $$i=1,2$$ i = 1 , 2 , the subgraph of G induced by $$V_i$$ V i is triangle-free and every vertex $$v\in V_i$$ v ∈ V i also has at least one neighbor in $$V_i$$ 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.