Uniqueness of rectangularly dualizable graphs

Document Type : Original paper

Authors

Department of Mathematics, BITS Pilani, Pilani Campus, Rajasthan - 333031

Abstract

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.

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