Co-even Domination Number of a Modified Graph by Operations on a Vertex or an Edge

TL;DR

This paper investigates the co-even domination number in graphs modified through operations on vertices or edges, providing insights into how such modifications affect this specific domination parameter.

Contribution

It introduces the concept of co-even domination number and analyzes its behavior under various graph operations involving vertices and edges.

Findings

Characterization of co-even domination number after vertex operations

Analysis of co-even domination number following edge modifications

Results on bounds and exact values for specific graph classes

Abstract

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $D\subseteq V$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is the domination number of $G$. A dominating set $D$ is called co-even dominating set if the degree of vertex $v$ is even number for all $v\in V-D$. The cardinality of a smallest co-even dominating set of $G$, denoted by $\gamma _{coe}(G)$, is the co-even domination number of $G$. In this paper we study co-even domination number of graphs which constructed by some operations on a vertex or an edge of a graph.