Independent domination in subcubic graphs

Abstract

A set S of vertices in a graph G is a dominating set if every vertex not in S is adjacent to a vertex in S. If, in addition, S is an independent set, then S is an independent dominating set. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set in G. In Goddard and Henning (Discrete Math 313:839–854, 2013) conjectured that if G is a connected cubic graph of order n, then $i(G) \le \frac{3}{8}n$, except if G is the complete bipartite graph $K_{3,3}$ or the 5-prism $C_5 \, \Box \, K_2$. Further they construct two infinite families of connected cubic graphs with independent domination three-eighths their order. In this paper, we provide a new family of connected cubic graphs G of order n such that $i(G) = \frac{3}{8}n$. We also show that if G is a subcubic graph of order n with no isolated vertex, then $i(G) \le \frac{1}{2}n$, and we characterize the graphs achieving equality in this bound.