The domination number of a graph G $G$, denoted γ ( 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 − 1 2 k − 1 false| V ( G ) false| $i(G)\le \frac{k-1}{2k-1}|V(G)|$ for a connected k $k$‐regular graph G $G$ that is not K k , 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 ) γ ( G ) ≤ k 3 − 3 k 2 + 2 2 k 2 − 6 k + 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 K k , k ${K}_{k,k}$, strengthening upon a result of Knor, Škrekovski, and Tepeh. In addition, we prove that a graph G $G$ with maximum degree at most 4 satisfies i ( G ) ≤ 5 9 false| V ( G ) false| $i(G)\le \frac{5}{9}|V(G)|$, which is also tight.

Given a graph , a dominating set of is a set of vertices such that each vertex not in has a neighbor in . Let denote the minimum size of a dominating set of . The independent domination number of , denoted , is the minimum size of a dominating set of that is also independent. We prove that if is a cubic graph without 4‐cycles, then , and the bound is tight. This result improves upon two results from two papers by Abrishami and Henning. Our result also implies that every cubic graph without 4‐cycles satisfies , which supports a question asked by O and West.