Co-even domination number of some binary operations on graphs

TL;DR

This paper investigates the co-even domination number, a graph invariant related to dominating sets with even degrees, specifically analyzing how it behaves under various binary graph operations.

Contribution

It introduces the concept of co-even domination number and explores its properties and behavior under different binary operations on graphs.

Findings

Derived bounds for co-even domination number under graph operations

Established relationships between co-even and standard domination numbers

Provided examples illustrating the effects of operations on co-even domination

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 the co-even domination number of some binary operations on graphs.