Cubic Graphs with Large Ratio of Independent Domination Number to Domination Number

Abstract

A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex outside $S$ has a neighbor in $S$; the domination number $\gamma (G)$ is the minimum size of such a set. The independent domination number, written $i(G)$, is the minimum size of a dominating set that also induces no edges. Henning and Southey conjectured $i(G)/\gamma (G) \le 6/5$ for every cubic (3-regular) graph $G$ with sufficiently many vertices. We provide an infinite family of counterexamples, giving for each positive integer $k$ a 2-connected cubic graph $H_k$ with $14k$ vertices such that $i(H_k)=5k$ and $\gamma (H_k)=4k$.