A tight bound for independent domination of cubic graphs without 4‐cycles

Abstract

Given a graph , a dominating set of is a set of vertices such that each vertex not in has a neighbor in . Let denote the minimum size of a dominating set of . The independent domination number of , denoted , is the minimum size of a dominating set of that is also independent. We prove that if is a cubic graph without 4‐cycles, then , and the bound is tight. This result improves upon two results from two papers by Abrishami and Henning. Our result also implies that every cubic graph without 4‐cycles satisfies , which supports a question asked by O and West.