Domination versus independent domination in regular graphs

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 $k\ge 3$ it holds that if $G$ is a connected $k$-regular graph, then $\frac{i(G)}{\gamma (G)}\le \frac{k}{2}$, with equality if and only if $G=K_{k,k}$. The result was previously known only for $k\le 6$. This affirmatively answers a recent question of Babikir and Henning.