Independent domination in subcubic graphs

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) <= 3/8n, except if G is the complete bipartite graph K-3,K-3 or the 5-prism C-5 square 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) = 3/8n. We also show that if G is a subcubic graph of order n with no isolated vertex, then i (G) <= 1/2n, and we characterize the graphs achieving equality in this bound.