Rigidity and reconstruction in matroids of highly connected graphs

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 Lov\'asz-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 Lov\'asz-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 Lov\'asz-Yemini (and thus the Whitney) property. These results unify and extend a number of earlier results about graph reconstruction from an underlying matroid.