On independent domination of regular graphs

TL;DR

This paper establishes new bounds on the independent domination number of regular graphs, generalizing previous results and answering open questions, with tight bounds demonstrated for specific cases.

Contribution

It proves a tight upper bound on the independent domination number for connected regular graphs not isomorphic to K_{k,k}, and strengthens bounds relating independent and domination numbers.

Findings

Bound i(G) ≤ ((k-1)/(2k-1))|V(G)| for regular graphs, tight at k=4.

Shows ratio i(G)/γ(G) ≤ (k^3-3k^2+2)/(2k^2-6k+2).

Establishes i(G') ≤ (5/9)|V(G')| for graphs with maximum degree 4, tight.

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. Note that every graph has an independent dominating set, as a maximal independent set is equivalent to an independent dominating set. Let $G$ be a connected $k$-regular graph that is not $K_{k, k}$ where $k\geq 4$. Generalizing a result by Lam, Shiu, and Sun, we prove that $i(G)\le \frac{k-1}{2k-1}|V(G)|$, which is tight for $k = 4$. This answers a question by Goddard et al. in the affirmative. We also show that $\frac{i(G)}{\gamma(G)} \le \frac{k^3-3k^2+2}{2k^2-6k+2}$, strengthening upon a result of Knor, \v{S}krekovski, and Tepeh. In addition, we prove that a graph $G'$ with maximum degree at most $4$ satisfies $i(G') \le \frac{5}{9}|V(G')|$, which is also tight.