A study on the complement graph of the completely separated topological graph

Articles in Press

Document Type : Original paper

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North Eastern Hill University Mawkynroh, Umshing, Shillong, India

Abstract

In this paper, we study $\overline{G(\tau)}$, the complement graph of the completely separated topological graph, and its line graph $L(\overline{G(\tau)})$ on a topological space $(X, \tau)$. We show that for a discrete topological space $(X, \tau)$, $\overline{G(\tau)}$ is Hamiltonian and Eulerian if and only if $|X|\geq 3$, and for any topological space $(X, \tau)$ such that $|X|\geq 3$, $e(X\backslash \{p\})=2$ for all $p \in X$ if and only if $(X,\tau)$ is a discrete space. Also, for any $T_1$ topological space $(X, \tau)$, $dt(\overline{G(\tau)})=2$ if and only if $X$ has at least one isolated point. Finally, if $(X, \tau_X)$ and $(Y, \tau_Y)$ are discrete topological spaces such that $|X|\geq 3$ and $|Y|\geq 3$, then $\overline{G(\tau_X)}$ is isomorphic to $\overline{G(\tau_Y)}$ if and only if $X$ and $Y$ are homeomorphic if and only if $L(\overline{G(\tau_X)})$ is isomorphic to $L(\overline{G(\tau_Y)})$.

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References

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Communications in Combinatorics and Optimization

Articles in Press, Accepted Manuscript
Available Online from 14 October 2025

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