1-MOVABLE DOUBLE OUTER-INDEPENDENT DOMINATION IN GRAPHS

Abstract

A nonempty set $S \subseteq V(G)$ is a 1-movable double outer-independent dominating set of $G$ if $S$ is a double outer-independent dominating set of $G$ and for every $v \in S, S \backslash\{v\}$ is a double outer-independent dominating set of $G$ or there exists a vertex $u \in(V(G) \backslash S) \cap N_G(v)$ such that $(S \backslash\{v\}) \cup\{u\}$ is a double outer-independent dominating set of $G$. The 1-movable double outer-independent domination number of a graph $G$, denoted by $\gamma_{m \times 2}^{1 o i}(G)$, is the smallest cardinality of a 1-movable double outer-independent dominating set of $G$. A 1-movable double outer-independent dominating set of $G$ with cardinality equal to $\gamma_{m \times 2}^{1 o i}(G)$ is called $\gamma_{m \times 2}^{1 o i}$-set of $G$. This paper characterizes 1-movable double outer-independent dominating sets in the join and corona of two graphs. Received: April 30, 2023 Accepted: July 1, 2023