Linear-Time Succinct Encodings of Planar Graphs via Canonical Orderings

TL;DR

This paper presents new linear-time encoding schemes for planar graphs that significantly reduce the number of bits needed, improving upon previous bounds for triangulated and triconnected graphs.

Contribution

It introduces succinct encoding methods for planar graphs using canonical orderings, achieving better bounds with linear-time encoding and decoding.

Findings

Triangulated graphs encoded with about 4/3 m bits, improving previous 1.53 m bound.

Triconnected graphs encoded with at most 2.835 m bits, better than 3 m bound.

Encoding and decoding processes run in linear time.

Abstract

Let G be an embedded planar undirected graph that has n vertices, m edges, and f faces but has no self-loop or multiple edge. If G is triangulated, we can encode it using {4/3}m-1 bits, improving on the best previous bound of about 1.53m bits. In case exponential time is acceptable, roughly 1.08m bits have been known to suffice. If G is triconnected, we use at most (2.5+2\log{3})\min\{n,f\}-7 bits, which is at most 2.835m bits and smaller than the best previous bound of 3m bits. Both of our schemes take O(n) time for encoding and decoding.