A Theory of Rectangularly Dualizable Graphs

TL;DR

This paper establishes a complete characterization of rectangularly dualizable graphs by providing necessary and sufficient conditions, correcting previous incomplete criteria, and expanding understanding of their structure in planar graph theory.

Contribution

It presents a new, comprehensive criterion for rectangularly dualizable graphs, correcting earlier conditions and including non-separable cases.

Findings

Counterexample showing previous conditions fail

New necessary and sufficient condition established

Enhanced understanding of rectangular graph duality

Abstract

A plane graph is called a rectangular graph if each of its edges can be oriented either horizontally or vertically, each of its interior regions is a four-sided region and all interior regions can be fitted in a rectangular enclosure. Only planar graphs can be dualized. If the dual of a plane graph is a rectangular graph, then the plane graph is a rectangularly dualizable graph. In 1985, Ko\'zmi\'nski and Kinnen presented a necessary and sufficient condition for the existence of a rectangularly dualizable graph for a separable connected plane graph. In this paper, we present a counter example for which the conditions given by them for separable connected plane graphs fail and hence, we derive a necessary and sufficient condition for a plane graph to be a rectangularly dualizable graph.