Forbidden minors for graphs with no first obstruction to parametric Feynman integration

TL;DR

This paper characterizes certain 3-connected planar graphs avoiding specific minors and relates these to graphs that do not obstruct parametric Feynman integration at the fifth step, providing a complete list of forbidden minors.

Contribution

It introduces a characterization of 3-connected planar graphs avoiding cube, octahedron, and a related minor, linking them to Feynman 5-split graphs and identifying all forbidden minors for these graphs.

Findings

3-connected planar graphs avoiding specific minors are characterized.

Feynman 5-split graphs are exactly those characterized above.

Complete list of forbidden minors for Feynman 5-split graphs of any connectivity.

Abstract

We give a characterization of 3-connected graphs which are planar and forbid cube, octahedron, and $H$ minors, where $H$ is the graph which is one $\Delta-Y$ away from each of the cube and the octahedron. Next we say a graph is Feynman 5-split if no choice of edge ordering gives an obstruction to parametric Feynman integration at the fifth step. The 3-connected Feynman 5-split graphs turn out to be precisely those characterized above. Finally we derive the full list of forbidden minors for Feynman 5-split graphs of any connectivity.