Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 Articles in Press 2025 10 14 A study on the complement graph of the completely separated topological graph 15027 10.22049/cco.2025.30478.2491 EN Pynshngain Dhar North Eastern Hill University Mawkynroh, Umshing, Shillong, India John Paul Jala Kharbhih North Eastern Hill University Mawkynroh, Umshing, Shillong, India Journal Article 2025 04 01 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 $L(\overline{G(\tau_Y)})$. https://comb-opt.azaruniv.ac.ir/article_15027_517eebe8c415df2da19bd7d9b2c366d2.pdf