Domination versus independent domination in cubic graphs

A set SS of vertices in a graph GG is a dominating set if every vertex not in SS is adjacent to a vertex in SS. If, in addition, SS is an independent set, then SS is an independent dominating set. The domination number γ(G)γ(G) of GG is the minimum cardinality of a dominating set in GG, while the independent domination number i(G)i(G) of GG is the minimum cardinality of an independent dominating set in GG. In this paper we show that if G≠K(3,3)G≠K(3,3) is a connected cubic graph, then i(G)/γ(G)≤4/3i(G)/γ(G)≤4/3. This answers a question posed in Goddard (in press) [6] where the bound of 3/23/2 is proven. In addition we characterize the graphs achieving this ratio of 4/34/3.