August 17, 2025

Domination Value in Graphs

Key Points

A dominating set of minimum cardinality is known as a γ(G)-set, and offers insight into graph structure.
For each vertex in a graph, the domination value indicates how many γ(G)-sets it belongs to, highlighting local properties.
Key findings include characterizations of domination value for specific graph types such as cycles and paths.
This research enhances understanding of edge connections and interactions in graph theory, promoting further exploration.

Abstract

A set D ⊆ V (G) is a dominating set of G if every vertex not in D is adjacent to at least one vertex in D. A dominating set of G of minimum cardinality is called a γ(G)-set. For each vertex v ∈ V (G), we define the domination value of v to be the number of γ(G)-sets to which v belongs. In this paper, we study some basic properties of the domination value function, thus initiating a local study of domination in graphs. Further, we characterize domination value for the Petersen graph, complete n-partite graphs, cycles, and paths.

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Cite This Study

Eunjeong Yi (Mon,) studied this question.

synapsesocial.com/papers/68a370e80a429f7973333400 https://doi.org/https://doi.org/10.55016/ojs/cdm.v7i2.62089

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