On independent domination of regular graphs

The domination number of a graph G $G$, denoted gamma(G) $\gamma (G)$, is the minimum size of a dominating set of G $G$, and the independent domination number of G $G$, denoted i(G) $i(G)$, is the minimum size of a dominating set of G $G$ that is also independent. Let k >= 4 $k\ge 4$ be an integer. Generalizing a result on cubic graphs by Lam, Shiu, and Sun, we prove that i(G)<= k-12k-1|V(G)| $i(G)\le \frac{k-1}{2k-1}|V(G)|$ for a connected k $k$-regular graph G $G$ that is not Kk,k ${K}_{k,k}$, which is tight for k=4 $k=4$. This answers a question by Goddard et al. in the affirmative. We also show that i(G)gamma(G)<= k3-3k2+22k2-6k+2 $\frac{i(G)}{\gamma (G)}\le \frac{{k}<^>{3}-3{k}<^>{2}+2}{2{k}<^>{2}-6k+2}$ for a connected k $k$-regular graph G $G$ that is not Kk,k ${K}_{k,k}$, strengthening upon a result of Knor, Skrekovski, and Tepeh. In addition, we prove that a graph G $G$ with maximum degree at most 4 satisfies i(G)<= 59|V(G)| $i(G)\le \frac{5}{9}|V(G)|$, which is also tight.