An excluded minor theorem for the 6-wheel

TL;DR

This paper completes the characterization of 3-connected $W_6$-minor-free graphs by analyzing the nonplanar cases, building on previous work that covered the planar instances.

Contribution

It extends Gubser's characterization by including the nonplanar 3-connected $W_6$-minor-free graphs.

Findings

Characterization of all 3-connected nonplanar $W_6$-minor-free graphs.

Complete classification of $W_6$-minor-free graphs.

Extension of existing planar graph results.

Abstract

For each integer $n \geq 3$, the wheel graph $W_n$ is defined as the graph obtained by connecting a single vertex to all vertices of a cycle of length $n$. In particular, $W_6$ can be uniquely obtained from the Petersen graph by contracting three edges incident to a common vertex. Gubser provided a characterization of all 3-connected planar $W_6$-minor-free graphs. In this paper, we complete the characterization of $W_6$-minor-free graphs by determining the 3-connected nonplanar cases.