A characterization of 4-connected graphs with no 6-wheel minor

TL;DR

This paper characterizes all 4-connected graphs that do not contain the 6-wheel graph as a minor, providing a complete structural understanding of such graphs.

Contribution

It provides a complete characterization of 4-connected graphs excluding the 6-wheel minor, expanding the understanding of graph minors and connectivity.

Findings

Identifies all 4-connected graphs without a W6 minor.

Establishes structural properties of these graphs.

Connects the absence of W6 minors to graph connectivity.

Abstract

For each integer $n\geq 3$, let $W_n$ denote the wheel graph obtained by connecting a single vertex to all vertices of a cycle of length $n$. In particular, $W_6$ is obtained from the Petersen graph by contracting three edges incident with a common vertex. In this paper, we determine all $4$-connected graphs that do not contain $W_6$ as a minor.