Excluding Pinched Spheres

TL;DR

This paper characterizes graphs excluding a specific pinched sphere-embeddable minor, providing a structural decomposition related to bidimensionality and embeddability in the projective plane, with implications for edge-apex graphs.

Contribution

It introduces a structural characterization of graphs excluding a pinched sphere minor, linking bidimensionality, tree decompositions, and embeddability in the projective plane.

Findings

Graphs excluding an $ ext{S}^ ext{o}_0$-embeddable minor have bounded bidimensionality in their tree decompositions.

Such graphs can be reduced to graphs embeddable in the projective plane by vertex identification.

Edge-apex graphs are characterized as graphs embeddable in the pinched sphere.

Abstract

The pinched sphere is the pseudo-surface $\mathbb{S}^{\circ}_0$ obtained by identifying two distinct points of the sphere. We provide a structural characterization of graphs excluding an $\mathbb{S}^{\circ}_0$-embeddable graph as a minor. Given a graph $G$ and a vertex set $X$, the bidimensionality of $X$ in $G$ is the maximum $k$ such that $G$ contains the $(k\times k)$-grid as an $X$-rooted minor, i.e., there exists a minor model of the $(k \times k)$-grid in~$G$ such that every branchset of this model contains a vertex of $X$. We prove that there is a function~$f$ such that, if a graph $G$ excludes an $\mathbb{S}^{\circ}_0$-embeddable graph $H$ as a minor, $G$ has a tree decomposition where each torso $G_{t}$ contains some set of vertices $X,$ whose bidimensionality in $G_{t}$ is at most $f(k)$ such that $G_{t}$ can be reduced to a graph embeddable in the projective plane by identifying vertices from $X$. This result is optimal in the sense that every graph admitting such a tree decomposition must exclude some $\mathbb{S}^{\circ}_0$-embeddable graph as a minor. An alternative interpretation of this result can be obtained by the fact that edge-apex graphs, i.e., graphs that can be made planar by removing an edge, are graphs embeddable in the pinched sphere. Several consequences and variants of this min-max duality are discussed.