Embeddings of *-graphs into 2-surfaces

TL;DR

This paper investigates embeddings of *-graphs with vertices of degree 4 or 6 into orientable 2-surfaces, providing an algorithm to determine planar embeddability based on matrix problems.

Contribution

It extends previous results on four-valent framed graphs to *-graphs with degree 4 or 6, establishing a quadratic-time algorithm for embedding into the plane.

Findings

Provides a matrix-based criterion for embeddings

Extends embedding algorithms to *-graphs with degree 4 or 6

Offers a quadratic-time algorithm for planarity testing

Abstract

This paper considers *-graphs in which all vertices have degree 4 or 6, and studies the question of calculating the genus of orientable 2-surfaces into which such graphs may be embedded. A *-graph is a graph endowed with a formal adjacency structure on the half-edges around each vertex, and an embedding of a *-graph is an embedding under which the formal adjacency relation on half-edges corresponds to the adjacency relation induced by the embedding. *-graphs are a natural generalization of four-valent framed graphs, which are four-valent graphs with an opposite half-edge structure. In [5], the question of whether a four-valent framed graph admits a Z2-homologically trivial embedding into a given surface was shown to be equivalent to a problem on matrices. We show that a similar result holds for *-graphs in which all vertices have degree 4 or 6. This gives an algorithm in quadratic time to determine whether a *-graph admits an embedding into the plane.