New classification of graphs in view of the domination number of central graphs

TL;DR

This paper introduces a new classification of graphs based on the relationship between the domination number of their central graphs and the minimum vertex cover, providing new insights into graph properties.

Contribution

It presents a novel classification of graphs into two categories based on domination and vertex cover numbers of central graphs, with additional new results on these parameters.

Findings

Graphs with at least three vertices are classified into two classes: those with γ(C(G))=τ(G) and those with γ(C(G))=τ(G)+1.

The paper establishes properties of vertex covers related to this classification.

New bounds and relationships between domination numbers and vertex covers of central graphs are provided.

Abstract

For a graph $G$, the central graph $C(G)$ is the graph constructed from $G$ by subdividing each edge of $G$ with one vertex and also by adding an edge to every pair of non-adjacent vertices in $G$. Also for a graph $G$, let $\gamma(G)$ and $\tau(G)$ be the domination number of $G$ and the minimum cardinarity of a vertex cover of $G$, respectively. In this paper, we give a new classification of graphs concerning the domination number of central graphs and minimum vertex covers of graphs. Namely, we show that any graph $G$ with at least three vertices can be classified into one of the two classes of graphs with $\gamma(C(G))=\tau(G)$ and $\gamma(C(G))=\tau(G)+1$, respectively, together with some special properties concerning a vertex cover of $G$. We also give some new results on the domination number of central graphs.