1-MOVABLE DOUBLE DOMINATION IN SOME BINARY OPERATIONS OF GRAPHS

Abstract

A double dominating set $D$ in a connected nontrivial graph $G$ without isolated vertices is a 1-movable double dominating set of $G$ if for every $v \in D$, either $D \backslash\{v\}$ is a double dominating set, or there exists a vertex $u \in(V(G) \backslash D) \cap N(v)$ such that $(D \backslash\{v\}) \cup\{u\}$ is a double dominating set of $G$. The minimum cardinality of a 1 -movable double dominating set of $G$, denoted by $\gamma_{m \times 2}^1(G)$, is the 1 -movable double domination number of $G$. A 1-movable double dominating set with cardinality $\gamma_{m \times 2}^1(G)$ is called a $\gamma_{m \times 2}^1$-set of $G$. In this paper, we characterize those graphs $G$ which possess a 1-movable double dominating set. We also characterize the 1-movable double dominating sets in the join and corona of graphs and determine the corresponding 1-movable double domination numbers of these graphs.

Received: December 28, 2022;
Accepted: February 18, 2023;