Orderly Spanning Trees with Applications

TL;DR

This paper introduces orderly spanning trees for plane graphs, providing a versatile tool with applications in graph drawing, encoding, and theoretical proofs, all achieved through linear-time algorithms.

Contribution

It generalizes canonical orderings to all connected planar graphs via orderly spanning trees and demonstrates their applications in graph visualization and encoding.

Findings

Linear-time algorithms for computing orderly pairs for any connected planar graph.

New constructive proof of Schnyder's Realizer Theorem using orderly spanning trees.

First area-optimal 2-visibility drawing of planar graphs.

Abstract

We introduce and study the {\em orderly spanning trees} of plane graphs. This algorithmic tool generalizes {\em canonical orderings}, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an {\em orderly pair} for any connected planar graph $G$, consisting of a plane graph $H$ of $G$, and an orderly spanning tree of $H$. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder's Realizer Theorem, (2) the first area-optimal 2-visibility drawing of $G$, and (3) the best known encodings of $G$ with O(1)-time query support. All algorithms in this paper run in linear time.