Tight bound for independent domination of cubic graphs without $4$-cycles

Abstract

Given a graph G, a dominating set of G is a set S of vertices such that each vertex not in S has a neighbor in S. The domination number of G, denoted γ(G), is the minimum size of a dominating set of G. The independent domination number of G, denoted i(G), is the minimum size of a dominating set of G that is also independent.Recently, Abrishami and Henning proved that if G is a cubic graph with girth at least 6, then i(G) ≤ 4 11 |V (G)|. We show a result that not only improves upon the upper bound of the aforementioned result, but also applies to a larger class of graphs, and is also tight. Namely, we prove that if G is a cubic graph without 4-cycles, then i(G) ≤ 5 14 |V (G)|, which is tight. Our result also implies that every cubic graph G without 4-cycles satisfies i(G) γ(G) ≤ 5 4 , which partially answers a question by O and West in the affirmative.