Domination versus independent domination in regular graphs

TL;DR

This paper establishes an upper bound on the ratio of independent domination number to domination number in connected regular graphs, confirming a conjecture for all regular graphs with degree at least 3.

Contribution

It proves that for all connected k-regular graphs with k ≥ 3, the ratio i(G)/γ(G) is at most k/2, with equality only for complete bipartite graphs, extending previous results.

Findings

The ratio i(G)/γ(G) ≤ k/2 for all connected k-regular graphs with k ≥ 3.

Equality holds only for the complete bipartite graph K_{k,k}.

The result confirms a conjecture for all k ≥ 3, previously known only for k ≤ 6.

Abstract

A set $S$ of vertices in a graph $G$ is a dominating set if every vertex of $G$ is in $S$ or is adjacent to a vertex in $S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The domination number $\gamma(G)$ of $G$ is the minimum cardinality of a dominating set in $G$, while the independent domination number $i(G)$ of $G$ is the minimum cardinality of an independent dominating set in $G$. We prove that for all integers $k \geq 3$ it holds that if $G$ is a connected $k$-regular graph, then $\frac{i(G)}{\gamma(G)} \leq \frac{k}{2}$, with equality if and only if $G = K_{k,k}$. The result was previously known only for $k\leq 6$. This affirmatively answers a recent question of Babikir and Henning.