Embedding simply connected 2-complexes in 3-space -- III. Constraint minors

TL;DR

This paper characterizes when a graphic matroid with a specified set of elements can be realized as a cycle matroid of a graph where that set forms a connected edge subset, using six obstructions.

Contribution

It introduces a new characterization involving six obstructions for embedding certain matroids with connectivity constraints.

Findings

Six obstructions characterize the property

Provides criteria for embedding graphic matroids with connected edge sets

Advances understanding of matroid and graph embedding relationships

Abstract

We characterise the following property by six obstructions: given a graphic matroid $M$ and a set $X$ of its elements, when is $M$ the cycle matroid of a graph $G$ such that $X$ is a connected edge set in $G$?