On Planarity of Graphs in Homotopy Type Theory

TL;DR

This paper develops a constructive, proof-relevant framework for graph planarity within homotopy type theory, providing an elementary topological characterization and formalising it in Agda.

Contribution

It introduces a novel homotopy type theoretic approach to graph planarity, including a formalisation in Agda, and characterizes planarity via maps and homotopy classes.

Findings

Planar graphs are characterised by maps with an outer face in homotopy type theory.

The set of planar maps forms a homotopy set for a graph.

Constructive methods for inductively building planar graphs are provided.

Abstract

In this paper, we present a constructive and proof-relevant development of graph theory, including the notion of maps, their faces, and maps of graphs embedded in the sphere, in homotopy type theory. This allows us to provide an elementary characterisation of planarity for locally directed finite and connected multigraphs that takes inspiration from topological graph theory, particularly from combinatorial embeddings of graphs into surfaces. A graph is planar if it has a map and an outer face with which any walk in the embedded graph is walk-homotopic to another. A result is that this type of planar maps forms a homotopy set for a graph. As a way to construct examples of planar graphs inductively, extensions of planar maps are introduced. We formalise the essential parts of this work in the proof-assistant Agda with support for homotopy type theory.