An excluded minor theorem for the Wagner graph plus an edge

TL;DR

This paper characterizes all internally 4-connected graphs that do not contain the graph $V_{8}+e$, which is formed by adding an edge to the Wagner graph, extending previous work on $V_{8}$ minors.

Contribution

It provides a complete characterization of internally 4-connected graphs excluding the $V_{8}+e$ minor, advancing the understanding of graph minors related to the Wagner graph.

Findings

Complete characterization of internally 4-connected $V_{8}+e$-minor-free graphs.

Extension of previous $V_{8}$ minor exclusion results.

New structural insights into graphs related to the Wagner graph.

Abstract

Let $V_{8}+e$ denote the unique graph obtained from the Wagner graph, also known as $V_{8}$, by adding an edge between two vertices of distance 3 on the Hamilton cycle, which is exactly a split of a minor of the Petersen graph. A complete characterization of all internally 4-connected graphs with no $V_{8}$ minor is given in [J. Maharry and N. Robertson, The structure of graphs not topologically containing the Wagner graph, J. Combin. Theory Ser. B 121 (2016) 398-420]. In this paper we characterize all internally 4-connected graphs with no $V_{8}+e$ minor.