Structural aspects of $1$-limited domination in cubic graphs

Abstract

A set $D\subseteq V(G)$ is a $1$-limited dominating set of a graph $G$ if every vertex outside $D$ has a neighbor in $D$ and every vertex of $D$ has at most one neighbor outside $D$. The minimum cardinality among all $1$-limited dominating sets of $G$ is denoted by $\gamma_1^{\mathrm L}(G)$. For every cubic graph $G$ of order $n$, there exists a nonnegative integer $t$ such that $\gamma_1^{\mathrm L}(G)=\frac n2+t$. We establish structural properties of $1$-limited dominating sets in terms of $t$ and use them to study the induced cycles vertex number, that is, the maximum number of vertices inducing a $2$-regular subgraph. As applications, we determine the exact $1$-limited domination numbers of generalized Blanu\v{s}a snarks and flower snarks.