A Whitney type theorem for surfaces: characterising graphs with locally planar embeddings

TL;DR

This paper extends Whitney's classical planar duality theorem to characterize graphs with locally planar embeddings on surfaces using matroid theory, providing a new criterion for local surface embeddability.

Contribution

It introduces a matroid-based characterization for r-locally planar embeddings of graphs on surfaces, generalizing Whitney's duality theorem.

Findings

r-locally 2-connected graphs embed r-locally in surfaces if and only if associated matroid is co-graphic

Extends Whitney's 1932 planar duality theorem to local surface embeddings

Provides a new matroid-theoretic criterion for graph embeddings on surfaces

Abstract

We prove that for any parameter r an r-locally 2-connected graph G embeds r-locally planarly in a surface if and only if a certain matroid associated to the graph G is co-graphic. This extends Whitney's abstract planar duality theorem from 1932.