Conflict graphs of maximally planar subgraphs of Petersen family graphs

TL;DR

This paper proves that most maximally planar subgraphs of Petersen Family graphs have unbalanced conflict graphs, with only three exceptions from a specific graph, K_{4,4}-e.

Contribution

It demonstrates that nearly all maximally planar subgraphs of Petersen Family graphs have unbalanced conflict graphs, identifying the unique balanced cases.

Findings

All but three conflict graphs are unbalanced.

The three balanced conflict graphs originate from K_{4,4}-e.

Most maximally planar subgraphs have unbalanced conflict graphs.

Abstract

The purpose of this paper is to show that all maximally planar subgraphs of graphs in the Petersen Family have associated conflict graphs unbalanced. All but three strong conflict graphs arising from Petersen Family Graphs are unbalanced, and the three that are balanced all come from $K_{4,4}-e$.