Rigidity and reconstruction in matroids of highly connected graphs

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.

💡 Research Summary

The paper investigates reconstruction of highly connected graphs from matroids that are defined uniformly on the edge set of every finite graph. A “graph matroid family” 𝓜 assigns to each graph G a matroid 𝓜(G) on its edges, with two natural requirements: (i) isomorphism invariance – graph isomorphisms induce matroid isomorphisms, and (ii) compatibility – the matroid of a subgraph is the restriction of the matroid of the larger graph. Within this framework the authors define two central properties.

The Whitney property states that there exists a constant c such that every c‑connected graph G is uniquely determined (up to isomorphism) by its matroid 𝓜(G). This generalises Whitney’s classic result that 3‑connected graphs are uniquely determined by their graphic matroids.

The Lovász‑Yemini property requires a constant c such that every c‑connected graph G has maximal possible rank among all graphs on the same vertex set; i.e., 𝓜(G) is “𝓜‑rigid”. This property was originally observed for rigidity matroids (Lovász‑Yemini proved that every 6‑connected graph is 2‑dimensional rigid).

A key dichotomy is introduced: a family 𝓜 is unbounded if the rank of 𝓜(Kₙ) grows without bound as n increases; otherwise it is bounded. For non‑trivial families the authors define two numerical invariants: the dimensionality d, the smallest integer for which an 𝓜‑circuit has minimum degree d+1, and the threshold t, the smallest vertex count minus one of such a circuit. Lemma 2.1 shows that for any n ≥ t, the rank satisfies
 r(Kₙ) = d·(n−t) + r(K_t),
so the rank grows linearly with slope d. When d = 0 the family is bounded; when d > 0 it is unbounded.

The main result, Theorem 1.1, proves that for unbounded families the Whitney and Lovász‑Yemini properties are equivalent. The forward direction (Whitney ⇒ Lovász‑Yemini) is straightforward (Lemma 3.2). The reverse direction is more involved: assuming the Lovász‑Yemini property with constant c, the authors construct a Whitney constant
 c′ = max(c, t) + rank(𝓜(K_{max(c,t)−1})) + 2,
where t is the threshold. The proof uses the Gluing Lemma (Lemma 2.3) to combine 𝓜‑rigid subgraphs and the linear rank formula to control the rank of added vertices.

In the bounded case every family automatically satisfies the Lovász‑Yemini property (Lemma 3.13) because the rank is capped, but not all bounded families have the Whitney property. The authors give a complete combinatorial characterization (Theorem 3.17) describing exactly which bounded families enjoy unique reconstruction.

Theorem 1.2 shows that taking unions of families preserves both properties: if each 𝓜_i has the Lovász‑Yemini (or Whitney) property, then so does the union 𝓜 = ⋃ 𝓜_i. This unifies many earlier results concerning graphic, bicircular, and (k,ℓ)-count matroids.

A further sufficient condition is introduced via 1‑extendability. A family is 1‑extendable if its dimensionality d is finite and a d‑dimensional edge‑split operation (replacing an edge by a new vertex adjacent to its endpoints and to d−1 additional vertices) preserves 𝓜‑independence. Theorem 1.3 proves that every 1‑extendable family has the Lovász‑Yemini property, and consequently the Whitney property. This captures known families such as k‑fold unions of graphic matroids, 2‑dimensional rigidity matroids, and the generic C_d‑2d‑1‑cofactor matroids, and extends to all abstract rigidity matroids that are 1‑extendable.

The paper also develops foundational lemmas: Lemma 2.2 characterises dimensionality via vertex‑addition, Lemma 2.3 (the Gluing Lemma) shows how to merge 𝓜‑rigid subgraphs, and Lemma 2.4 describes the rank of bounded families and the structure of minimal dependent sets.

In the concluding section the authors outline open problems: whether the converse of Theorem 1.3 holds, a tighter bound for the Whitney constant c′, algorithmic aspects of reconstruction (beyond existence), and a deeper combinatorial description of bounded families with the Whitney property.

Overall, the work provides a unified framework for graph reconstruction from matroids, clarifies the precise relationship between maximal rank and unique reconstruction, and introduces new tools—dimensionality, threshold, and 1‑extendability—that both generalise and simplify many earlier results in rigidity theory and matroid‑based graph reconstruction.