Sufficient conditions for bipartite rigidity, symmetric completability and hyperconnectivity of graphs
Dániel Garamvölgyi, Bill Jackson, Tibor Jordán, Soma Villányi
TL;DR
The paper studies Kalai's three matroid families on graph edge sets—symmetric completion $\mathcal{S}_d$, hyperconnectivity $\mathcal{H}_d$, and birigidity $\mathcal{B}_d$—and links their ranks to low-rank matrix completion problems. It establishes sufficiency conditions for reaching maximum matroid rank: high minimum degree suffices for $\mathcal{S}_d$ and $\mathcal{H}_d$ while high connectivity suffices for $\mathcal{B}_d$, employing novel seed-based techniques, extension operations, and a new notion of $k$-biconnectivity. The authors prove almost-tight bounds $h_d,s_d=O(d^2)$ and show that $k_d=O(d^3)$ ensures $d$-birigidity in bipartite graphs, with extensions to $(a,b)$-birigidity. They also derive a vertex-cover lower bound in critically connected bipartite graphs and develop probabilistic seed constructions, advancing the combinatorial rigidity framework for matrix completion problems. These results provide structural criteria for when partial entries determine full-rank completions in various matrix symmetry contexts, with implications for unique recoverability in low-rank matrix problems.
Abstract
We consider three matroids defined by Kalai in 1985: the symmetric completion matroid $\mathcal{S}_d$ on the edge set of a looped complete graph; the hyperconnectivity matroid $\mathcal{H}_d$ on the edge set of a complete graph; and the birigidity matroid $\mathcal{B}_d$ on the edge set of a complete bipartite graph. These matroids arise in the study of low rank completion of partially filled symmetric, skew-symmetric and rectangular matrices, respectively. We give sufficient conditions for a graph $G$ to have maximum possible rank in these matroids. For $\mathcal{S}_d$ and $\mathcal{H}_d$, our conditions are in terms of the minimum degree of $G$ and are best possible. For $\mathcal{B}_d$, our condition is in terms of the connectivity of $G$. Our results are analogous to recent results for rigidity matroids due to Krivelevich, Lew and Michaeli, and Villányi, respectively, but our proofs require new techniques and structural results. In particular, we give an almost tight lower bound on the vertex cover number in critically $k$-connected graphs.