Domination versus independent domination in cubic graphs

A set S of vertices in a graph G is a dominating set if every vertex not in S 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 γ ( 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 . In this paper we show that if G ≠ K ( 3 , 3 ) is a connected cubic graph, then i ( G ) / γ ( G ) ≤ 4 / 3 . This answers a question posed in Goddard (in press) [6] where the bound of 3 / 2 is proven. In addition we characterize the graphs achieving this ratio of  4 / 3 .