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

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