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

Given a graph G $G$, a dominating set of G $G$ is a set S $S$ of vertices such that each vertex not in S $S$ has a neighbor in S $S$. Let gamma(G) $\gamma (G)$ denote the minimum size of a dominating set of G $G$. The independent domination number of G $G$, denoted i(G) $i(G)$, is the minimum size of a dominating set of G $G$ that is also independent. We prove that if G $G$ is a cubic graph without 4-cycles, then i(G)<= 514 divide V(G) divide $i(G)\le \frac{5}{14}| V(G)| $, 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 G $G$ without 4-cycles satisfies i(G)gamma(G)<= 54 $\frac{i(G)}{\gamma (G)}\le \frac{5}{4}$, which supports a question asked by O and West.