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arXiv · 2411.07347

An Efficient Genus Algorithm Based on Graph Rotations

Alexander Metzger·Austin Ulrigg
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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.

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Alexander Metzger, Austin Ulrigg. 2024-11-11. An Efficient Genus Algorithm Based on Graph Rotations. https://doi.org/10.1016/j.disc.2026.115308

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