Ranking and Unranking of the Planar Embeddings of a Planar Graph

TL;DR

This paper introduces an efficient bijection between all planar embeddings of a graph and natural numbers, enabling fast ranking, unranking, and uniform random generation of embeddings for practical applications.

Contribution

It presents a novel linear-time ranking and near-linear-time unranking method for all planar embeddings of a graph, facilitating enumeration and random sampling.

Findings

Ranking function computed in O(n) time

Unranking function computed in O(n α(n)) time

Enables uniform random generation of planar embeddings

Abstract

Let $\mathcal{G}$ be the set of all the planar embeddings of a (not necessarily connected) $n$-vertex graph $G$. We present a bijection $\Phi$ from $\mathcal{G}$ to the natural numbers in the interval $[0 \dots |\mathcal{G}| - 1]$. Given a planar embedding $\mathcal{E}$ of $G$, we show that $\Phi(\mathcal{E})$ can be decomposed into a sequence of $O(n)$ natural numbers each describing a specific feature of $\mathcal{E}$. The function $\Phi$, which is a ranking function for $\mathcal{G}$, can be computed in $O(n)$ time, while its inverse unranking function $\Phi^{-1}$ can be computed in $O(n \alpha(n))$ time. The results of this paper can be of practical use to uniformly at random generating the planar embeddings of a graph $G$ or to enumerating such embeddings with amortized constant delay. Also, they can be used to counting, enumerating or uniformly at random generating constrained planar embeddings of $G$.