RESTRAINED CRITICAL AND ABUNDANT SIGNED GRAPHS

Abstract

Let $\Sigma=(V, E, \sigma)$, where $\sigma: E(|\Sigma|) \rightarrow\{+,-\}$ is a signed graph. A set $D \subseteq V$ is said to be a restrained dominating set of $\sum$, if $D$ is a restrained dominating set of and every cycle formed by the edges across $D$ to $V D$ and within $V D$ is balanced. The cardinality of a minimum restrained dominating set in $\Sigma$ is the restrained domination number of $\Sigma$ and is denoted by $\gamma_r(\Sigma)$. In this paper, we investigate how the addition of an edge between any two non-adjacent vertices or the removal of a vertex from any signed graph affects the restrained domination number for heterogeneous signed graphs.

Received: November 29, 2022;
Accepted: January 24, 2023;