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arXiv · 2204.10292

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

Shinya Fujita·Farshad Kazemnejad·Behnaz Pahlavsay
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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.

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Shinya Fujita, Farshad Kazemnejad, Behnaz Pahlavsay. 2022-04-20. New classification of graphs in view of the domination number of central graphs. https://arxiv.org/abs/2204.10292

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