Rigidity and reconstruction in matroids of highly connected graphs

TL;DR

The paper investigates how graphs can be reconstructed from graph matroid families $ ext{M}(G)$, introducing the Whitney and Lovász-Yemini properties as formal measures of reconstructibility and rigidity for highly connected graphs. It establishes a sharp dichotomy: for unbounded matroid families, the Whitney property is equivalent to the Lovász-Yemini property, with a quantitative bound linking the two via the threshold and rank on complete graphs; in the bounded case, it provides a complete combinatorial characterization of when Whitney holds. The authors develop a robust framework using dimensionality, threshold, and vertical connectivity to relate matroid structure to graph connectivity, and show that unions of matroid families preserve both properties while identifying a broad class of 1-extendable families that automatically satisfy LY (and Whitney). These results unify and extend prior results on graph reconstruction from underlying matroids, connect to rigidity matroids and abstract rigidity matroids, and open avenues for future exploration in broader combinatorial settings and reconstruction algorithms.

Abstract

A graph matroid family $\mathcal{M}$ is a family of matroids $\mathcal{M}(G)$ defined on the edge set of each finite graph $G$ in a compatible and isomorphism-invariant way. We say that $\mathcal{M}$ has the Whitney property if there is a constant $c$ such that every $c$-connected graph $G$ is uniquely determined by $\mathcal{M}(G)$. Similarly, $\mathcal{M}$ has the Lovász-Yemini property if there is a constant $c$ such that for every $c$-connected graph $G$, $\mathcal{M}(G)$ has maximal rank among graphs on the same number of vertices. We show that if $\mathcal{M}$ is unbounded (that is, there is no absolute constant bounding the rank of $\mathcal{M}(G)$ for every $G$), then $\mathcal{M}$ has the Whitney property if and only if it has the Lovász-Yemini property. We also give a complete characterization of these properties in the bounded case. As an application, we show that if some graph matroid families have the Whitney property, then so does their union. Finally, we show that every $1$-extendable graph matroid family has the Lovász-Yemini (and thus the Whitney) property. These results unify and extend a number of earlier results about graph reconstruction from an underlying matroid.

Theorems & Definitions (60)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof