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Uniqueness of rectangularly dualizable graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 2، دوره 9، شماره 1، خرداد 2024، صفحه 13-25 اصل مقاله (409.26 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2022.27774.1350 | ||
| نویسندگان | ||
| Vinod Kumar* ؛ Krishnendra Shekhawat | ||
| Department of Mathematics, BITS Pilani, Pilani Campus, Rajasthan - 333031 | ||
| چکیده | ||
| A generic rectangular partition is a partition of a rectangle into a finite number of rectangles provided that no four of them meet at a point. A graph $\mathcal{H}$ is called dual of a plane graph $\mathcal{G}$ if there is one$-$to$-$one correspondence between the vertices of $\mathcal{G}$ and the regions of $\mathcal{H}$, and two vertices of $\mathcal{G}$ are adjacent if and only if the corresponding regions of $\mathcal{H}$ are adjacent. A plane graph is a rectangularly dualizable graph if its dual can be embedded as a rectangular partition. A rectangular dual $\mathcal{R}$ of a plane graph $\mathcal{G}$ is a partition of a rectangle into $n-$rectangles such that (i) no four rectangles of $\mathcal{R}$ meet at a point, (ii) rectangles in $\mathcal{R}$ are mapped to vertices of $\mathcal{G}$, and (iii) two rectangles in $\mathcal{R}$ share a common boundary segment if and only if the corresponding vertices are adjacent in $\mathcal{G}$. In this paper, we derive a necessary and sufficient for a rectangularly dualizable graph $\mathcal{G}$ to admit a unique rectangular dual upto combinatorial equivalence. Further we show that $\mathcal{G}$ always admits a slicible as well as an area$-$universal rectangular dual. | ||
| کلیدواژهها | ||
| plane graphs؛ rectangularly dualizable graphs؛ rectangular duals؛ rectangular partitions | ||
| مراجع | ||
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