An Efficient Genus Algorithm Based on Graph Rotations

TL;DR

This paper introduces an efficient algorithm for computing the minimal genus of a graph, which is simple to implement, handles bridge placements well, and provides detailed embedding information.

Contribution

The paper presents a novel algorithm that determines the orientable genus of any graph with improved efficiency and practical features, addressing limitations of previous methods.

Findings

Successfully computed the genus of the (3,12) cage as 17

Algorithm effectively narrows bounds for genus estimation

Provides faces of an optimal embedding

Abstract

We study the problem of determining the minimal genus of a simple finite connected graph. We present an algorithm which, for an arbitrary graph $G$ with $n$ vertices and $m$ edges, determines the orientable genus of $G$ in $O(n(4^m/n)^{n/t})$ steps where $t$ is the girth of $G$. This algorithm avoids difficulties that many other genus algorithms have with handling bridge placements which is a well-known issue. The algorithm has a number of useful properties for practical use: it is simple to implement, it outputs the faces of an optimal embedding, and it iteratively narrows both upper and lower bounds. We illustrate the algorithm by determining the genus of the $(3,12)$ cage (which is 17); other graphs are also considered.