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 2013 Goddard and Henning [Discrete Math 313 (2013), 839-854] conjectured that if G is a connected cubic graph of order n, then i(G) ≤ 3 8 n, except if G is the complete bipartite graph K 3,3 or the 5-prism C 5 2 K 2 . Further they construct two infinite families of connected cubic graphs with independent domination three-eighths their order. They remark that perhaps it is even true that for n > 10 these two families are only families for which equality holds. In this paper, we provide a new family of connected cubic graphs G of order n such that i(G) = 3 8 n. We also show that if G is a subcubic graph of order n with no isolated vertex, then i(G) ≤ 1 2 n, and we characterize the graphs achieving equality in this bound.