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

TL;DR

This paper establishes a tight upper bound on the independent domination number for cubic graphs without 4-cycles, improving previous bounds and extending applicability to a broader class of graphs.

Contribution

It proves a new tight upper bound of 5/14 of the vertices for the independent domination number in such graphs, generalizing and strengthening prior results.

Findings

Bound of i(G) ≤ 5/14 |V(G)| for cubic graphs without 4-cycles

The ratio i(G)/γ(G) ≤ 5/4 for these graphs

The result is tight and partially answers a known question

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 $\gamma(G)$, 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 $i(G) \le \frac{4}{11}|V(G)|$. 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 $i(G) \le \frac{5}{14}|V(G)|$, which is tight. Our result also implies that every cubic graph $G$ without $4$-cycles satisfies $\frac{i(G)}{\gamma(G)} \le \frac{5}{4}$, which partially answers a question by O and West in the affirmative.