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 $\gamma(G)$, 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 $K_{k, k}$ where $k\geq 4$. Generalizing a result by Lam, Shiu, and Sun, we prove that $i(G)\le \frac{k-1}{2k-1}|V(G)|$, which is tight for $k = 4$. This answers a question by Goddard et al. in the affirmative. We also show that $\frac{i(G)}{\gamma(G)} \le \frac{k^3-3k^2+2}{2k^2-6k+2}$, strengthening upon a result of Knor, \v{S}krekovski, and Tepeh. In addition, we prove that a graph $G'$ with maximum degree at most $4$ satisfies $i(G') \le \frac{5}{9}|V(G')|$, which is also tight.Comment: 15 pages, 5 figure

Independent Domination in Subcubic 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 independent domination number $i(G)$ of $G$ is the minimum cardinality of an independent dominating set in $G$. In 2013 Goddard and Henning [Discrete Math 313 (2013), 839--854] conjectured that if $G$ is a connected cubic graph of order $n$, then $i(G) \le \frac{3}{8}n$, except if $G$ is the complete bipartite graph $K_{3,3}$ or the $5$-prism $C_5 \, \Box \, K_2$. Further they construct two infinite families of connected cubic graphs with independent domination three-eighths their order. They remark that perhaps it is even true that for $n > 10$ these two families are only families for which equality holds. In this paper, we provide a new family of connected cubic graphs $G$ of order $n$ such that $i(G) = \frac{3}{8}n$. We also show that if $G$ is a subcubic graph of order $n$ with no isolated vertex, then $i(G) \le \frac{1}{2}n$, and we characterize the graphs achieving equality in this bound.Comment: Submitted to Discrete Applied Mathematics Journal, 08 Jan 202