The (3,1)-ordering for 4-connected planar triangulations

Therese Biedl,Martin Derka

doi:10.7155/jgaa.00396

Abstract

Canonical orderings of planar graphs have frequently been used in graph drawing and other graph algorithms. In this paper we introduce the notion of an $(r,s)$-canonical order, which unifies many of the existing variants of canonical orderings. We then show that $(3,1)$-canonical ordering for 4-connected triangulations always exist; to our knowledge this variant of canonical ordering was not previously known. We use it to give much simpler proofs of two previously known graph drawing results for 4-connected planar triangulations, namely, rectangular duals and rectangle-of-influence drawings.

Highlights

A canonical ordering of a planar graph is a way of building the graph by iteratively attaching vertices to some “basic graph” while preserving some connectivity invariant after each iteration
It is substantially different from the canonical ordering for such graphs that was presented by Kant and He [KH97]
A rectangular dual drawing of a planar graph G consists of a set of interior-disjoint rectangles assigned to the vertices of G in such a way that the union of the rectangles forms a rectangle without holes, and the rectangles assigned to vertices v and w touch in a non-zero-length line segment if and only if (v, w) is an edge

Summary

Background

A canonical ordering of a planar graph is a way of building the graph by iteratively attaching vertices to some “basic graph” (such as an edge) while preserving some connectivity invariant after each iteration. This concept was introduced in the late 1980’s by de Fraysseix, Pach and Pollack [dFPP90]. (We will review these below.) In this paper, we show the existence yet another canonical ordering, this one for planar 4-connected triangulations It is substantially different from the canonical ordering for such graphs that was presented by Kant and He [KH97]. We use the (3, 1)-canonical ordering to provide alternate (and, in our opinion, significantly simpler) proofs of two previously known results about 4-connected planar triangulations: they have rectangular duals (Section 4.1) and rectangle-of-influence drawings (Section 4.2)

Review of existing canonical orderings

Rectangular duals

Rectangle-of-influence drawings

Conclusion