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A study on the complement graph of the completely separated topological graph | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 22 مهر 1404 اصل مقاله (413.44 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2025.30478.2491 | ||
| نویسندگان | ||
| Pynshngain Dhar* ؛ John Paul Jala Kharbhih | ||
| North Eastern Hill University Mawkynroh, Umshing, Shillong, India | ||
| چکیده | ||
| 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 $L(\overline{G(\tau_Y)})$. | ||
| کلیدواژهها | ||
| Open set؛ topological spaces؛ continuous function؛ complete graph؛ dominating set | ||
| مراجع | ||
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[2] P. Dhar and J.P.J. Kharbhih, Completely separated topological graph, Palest. J. Math. 14 (2025), no. 2, 531–539.
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[4] R.A. Muneshwar and K.L. Bondar, Open subset inclusion graph of a topological space, J. Discrete Math. Sci. Cryptogr. 22 (2019), no. 6, 1007–1018. https://doi.org/10.1080/09720529.2019.1649029
[5] S. Semmes, Some Basic Topics in Topology and Set Theory, Rice University.
[6] M. Venkatachalapathy, K. Kokila, and B. Abarna, Some trends in line graphs, Adv. Theo. Appl. Math. 11 (2016), no. 2, 171–178.
[7] D.B. West, Introduction to Graph Theory, Prentice-Hall Upper Saddle River, 2001.
[8] S. Willard, General Topology, Courier Corporation, 2012. | ||
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