Simple realizability of complete abstract topological graphs simplified

TL;DR

This paper characterizes simply realizable complete abstract topological graphs using a finite set of forbidden subgraphs, leading to a polynomial-time algorithm for testing realizability.

Contribution

It provides a finite forbidden subgraph characterization for simple realizability of complete AT-graphs, simplifying previous algorithms.

Findings

Finite forbidden AT-subgraph characterization with at most six vertices.

Polynomial-time algorithm for testing simple realizability.

Extension of results to $ ext{Z}_2$-realizability based on crossing parity.

Abstract

An abstract topological graph (briefly an AT-graph) is a pair $A=(G,\mathcal{X})$ where $G=(V,E)$ is a graph and $\mathcal{X}\subseteq {E \choose 2}$ is a set of pairs of its edges. The AT-graph $A$ is simply realizable if $G$ can be drawn in the plane so that each pair of edges from $\mathcal{X}$ crosses exactly once and no other pair crosses. We show that simply realizable complete AT-graphs are characterized by a finite set of forbidden AT-subgraphs, each with at most six vertices. This implies a straightforward polynomial algorithm for testing simple realizability of complete AT-graphs, which simplifies a previous algorithm by the author. We also show an analogous result for independent $\mathbb{Z}_2$-realizability, where only the parity of the number of crossings for each pair of independent edges is specified.