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 $\gamma_{coe}(G)$, defined as the minimum size of a co-even dominating set in $G$ where every vertex in $V-D$ has even degree. It develops tight two-sided bounds for $\gamma_{coe}$ under vertex removal $G-v$, edge removal $G-e$, vertex contraction $G/v$, and edge contraction $G/e$, and demonstrates the sharpness of these bounds via constructions and examples. An immediate relation linking $\gamma_{coe}(G)$ to the pair $(\gamma_{coe}(G-v), \gamma_{coe}(G/v))$ (and an analogous result for $G/e$) is established, highlighting how $\gamma_{coe}$ changes under common graph modifications. The results extend prior work on co-even domination and provide analytic tools for assessing stability of $\gamma_{coe}$ under graph operations, with potential guidance for studying other unary graph transformations.

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 $γ(G)$, 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 $γ_{coe}(G)$, 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.

Figures (10)

• Figure 1: Graphs $G$ and $G-v$
• Figure 2: Graphs $H$ and $H-v$
• Figure 3: Graphs $H$ and $H-v$ with same co-even domination numbers
• Figure 4: Graphs $G$ and $G-e$
• Figure 5: Graphs $H$ and $H-e$

Theorems & Definitions (18)

• Proposition 2.1
• Proposition 2.2
• Theorem 2.3
• proof
• Remark 2.4
• Example 2.5
• Theorem 2.6
• proof
• Remark 2.7
• Theorem 3.1