Transformations of Rectangular Dualizable Graphs

TL;DR

This paper studies the properties and transformations of rectangular dualizable graphs (RDGs), introducing polynomial algorithms for adjacency transformations, and characterizing maximal and minimal RDGs through edge-reducibility concepts.

Contribution

It presents polynomial algorithms for transforming RDGs, characterizes maximal and minimal RDGs, and proves the existence of maximal RDGs for any RDG.

Findings

Existence of a maximal RDG for any RDG.

Maximal RDGs are edge-reducible and can be transformed into minimal RDGs.

Polynomial time algorithms for adjacency transformations of RDGs.

Abstract

A plane graph is said to be a rectangular graph if each of its edges can be oriented horizontal or vertical, its internal regions are four-sided and it has a rectangular enclosure. If dual of a planar graph is a rectangular graph, then the graph is said to be a rectangular dualizable graph (RDG). In this paper, we present adjacency transformations between RDGs and present polynomial time algorithms for their transformations. An RDG $\mathcal{G}=(V, E)$ is called maximal RDG (MRDG) if there does not exist an RDG $\mathcal{G'}=(V, E')$ with $E' \supset E$. An RDG $\mathcal{G}=(V, E)$ is said to be an edge-reducible if there exists an RDG $\mathcal{G'}=(V, E')$ such that $E\supset E'$. If an RDG is not edge-reducible, it is said to be an edge-irreducible RDG. We show that there always exists an MRDG for a given RDG. We also show that an MRDG is edge-reducible and can always be transformed to a minimal one (an edge-irreducible RDG).