On independent domination of regular graphs

Given a graph G, a dominating set of G is a set S of vertices such that each vertex not in S has a neighbor in S. The domination number of G, denoted γ(G), is the minimum size of a dominating set of G. The independent domination number of G, denoted i(G), is the minimum size of a dominating set of G that is also independent. Note that every graph has an independent dominating set, as a maximal independent set is equivalent to an independent dominating set. Let G be a connected k-regular graph that is not Kk,k where k ≥ 4. Generalizing a result by Lam, Shiu, and Sun, we prove that i(G) ≤ k−1 2k−1 |V (G)|, which is tight for k = 4. This answers a question by Goddard et al. in the affirmative. We also show that i(G) γ(G) ≤ k 3−3k2+2 2k2−6k+2 , strengthening upon a result of Knor, Škrekovski, and Tepeh. In addition, we prove that a graph G′ with maximum degree at most 4 satisfies i(G′) ≤ 5 9 |V (G′)|, which is also tight.