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

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 γ(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 that if G is a connected cubic graph with sufficiently many vertices, then i(G)/γ(G) ≤ 6/5. We provide an infinite family of counterexamples, giving for each positive integer k a 2-connected cubic graph Hk with 14k vertices such that i(Hk) = 5k and γ(G) = 4k.