Graph Genera and Minors

Abstract

This thesis studies embedding graphs on surfaces, with an emphasis on orientable surfaces, rotation systems, and forbidden minors. We begin by developing the historical background, from the earliest occurrences of the crossing number $\operatorname{cr}(G)$ of a graph, to Heawood's map-coloring conjecture and the eventual invention of rotation systems as a way of encoding cellular embeddings of graphs on surfaces. We then describe and analyze \textsc{PAGE}, a rotation system based genus algorithm developed jointly with Alexander Metzger. For an arbitrary graph $G$ with $n$ vertices and $m$ edges, \textsc{PAGE} determines the orientable genus of $G$ in $\mathcal{O}(n(4^m/n)^{n/t})$ steps where $t$ is the \textit{girth} of $G$. We illustrate \textsc{PAGE} with examples where it was used to determine the genus of graphs whose genus was previously unknown, most notably the $(3,12)$ cage which has genus 17. Additionally, we discuss ways in which \textsc{PAGE} can be improved and other strategies that could be used to determine the genus of graphs such as the $(3,11)$ cage which are so far out of reach for algorithmic approaches. We also give an overview of the problem of determining the full set $\mathfrak{F}(S_1)$ of forbidden toroidal minors, summarize known results, and discuss possible directions for further progress on completing the set.