1-MOVABLE 2-OUTER-INDEPENDENT DOMINATING SETS IN GRAPHS

Abstract

A set $\varnothing \neq S \subseteq V(G)$ is a 1-movable 2-outer-independent dominating set of $G$ if $S$ is a 2-outer-independent dominating set of $G$ and for every $v \in S, S \backslash\{v\}$ is a 2-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 2 -outer-independent dominating set of $G$. The 1-movable 2 -outerindependent domination number of a graph $G$, denoted by $\gamma_{m 2}^{10 i}(G)$, is the smallest cardinality of a 1-movable 2-outer-independent dominating set of $G$. A 1 -movable 2 -outer-independent dominating set of $G$ with cardinality equal to $\gamma_{m 2}^{10 i}(G)$ is called $\gamma_{m 2}^{1 o i}$-set of $G$. We characterize 1-movable 2 -outer-independent dominating set in a graph and the 1-movable 2-outer-independent domination in some special graphs.

Received: April 10, 2024
Revised: September 28, 2024
Accepted: October 1, 2024