Uniqueness of rectangularly dualizable graphs

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