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

TL;DR

This paper introduces the (3,1)-canonical ordering for 4-connected triangulations, providing a new tool that simplifies existing graph drawing methods like rectangular duals and rectangle-of-influence drawings.

Contribution

It defines the (3,1)-canonical ordering and proves its existence for 4-connected triangulations, offering simpler proofs for key graph drawing results.

Findings

Existence of (3,1)-canonical ordering for 4-connected triangulations

Simplified proofs for rectangular duals

Simplified proofs for rectangle-of-influence drawings

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.