Domination versus independent domination in graphs of small regularity

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. It is known (Goddard et al., 2012) that if G is a connected 3-regular graph, then i(G)∕γ(G)≤3∕2, with equality if and only if G=K_3,3. In this paper, we extend this result to graphs of larger regularity and show that if k∈{4,5,6} and G is a connected k-regular graph, then i(G)∕γ(G)≤k∕2, with equality if and only if G=K_k,k.