Uniqueness of rectangularly dualizable graphs

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 H is called dual of a plane graph G if there is one−to−one correspondence between the vertices of G and the regions of H, and two vertices of G are adjacent if and only if the corresponding regions of H are adjacent. A plane graph is a rectangularly dualizable graph if its dual can be embedded as a rectangular partition. A rectangular dual R of a plane graph G is a partition of a rectangle into n−rectangles such that (i) no four rectangles of R meet at a point, (ii) rectangles in R are mapped to vertices of G, and (iii) two rectangles in R share a common boundary segment if and only if the corresponding vertices are adjacent in G. In this paper, we derive a necessary and sufficient for a rectangularly dualizable graph G to admit a unique rectangular dual upto combinatorial equivalence. Further we show that G always admits a slicible as well as an area−universal rectangular dual.