Sufficient conditions for bipartite rigidity, symmetric completability and hyperconnectivity of graphs

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 have several implications for the unique completability of low-rank matrices. In particular, they imply that: almost all sufficiently large $n \times n$ positive semidefinite matrices of rank $d$ are uniquely determined by any subset of their entries which includes at least $(n + d + 1)/2$ entries from each row; almost all $m \times n$ matrices of rank $d$ are uniquely determined by any subset of their entries whose positions define a spanning subgraph of $K{m,n}$ which is $k_d$-connected, for some constant $k_d=\mbox{O}(d^3)$.

💡 Research Summary

The paper studies three matroids introduced by Kalai in 1985: the symmetric completion matroid S_d defined on the edge set of a looped complete graph, the hyperconnectivity matroid H_d defined on the edge set of a simple complete graph, and the birigidity matroid B_d defined on the edge set of a complete bipartite graph. These matroids model low‑rank completion problems for symmetric, skew‑symmetric, and rectangular matrices, respectively. The authors give sufficient conditions under which a graph attains the maximum possible rank in each of these matroids.

For S_d and H_d, the conditions are expressed in terms of the minimum vertex degree. Theorem 1.2 shows that for any integer d ≥ 1 there exist constants h_d and s_d of order O(d²) such that (a) every simple graph on n ≥ h_d vertices with minimum degree at least (n + d − 1)/2 is d‑hyperconnected (i.e., its edge set spans H_d), and (b) every semisimple graph on n ≥ s_d vertices whose minimum degree is at least (n + d − 1)/2 and whose non‑loop vertices have degree at least (n + d)/2 is d‑completable (i.e., its edge set spans S_d). The authors prove that these degree bounds are best possible for d ≥ 2.

For the bipartite case, Theorem 1.3 establishes that there exists a constant k_d = O(d³) such that every k_d‑connected bipartite graph is d‑birigid (its edge set spans B_d). The paper also constructs examples of d²‑connected bipartite graphs that fail to be d‑birigid, showing that the connectivity requirement is within a factor of d of optimal.

The paper connects these graph‑theoretic results to matrix completion. Theorem 1.4 states two concrete implications: (a) a generic n × n positive semidefinite matrix of rank d is uniquely determined by any set of entries that includes at least (n + d + 1)/2 entries in each row; (b) a generic m × n matrix of rank d is uniquely determined by any set of entries whose positions form a spanning subgraph of K_{m,n} that is (k_d + 1)‑connected. These statements improve on earlier results by requiring far fewer observed entries for unique recovery.

To prove the main theorems, the authors develop new extension operations that preserve independence in the three matroids. While the classic 1‑extension operation works for the rigidity matroid R_d, it does not preserve independence in S_d, H_d, or B_d. The authors therefore introduce three operations: the 0‑extension, the double 1‑extension, and the looped 1‑extension, and prove that they preserve independence in the respective matroids (Lemma 2.1). For the bipartite case they also define a new notion of k‑biconnectivity, prove lower bounds on vertex‑cover numbers in critically k‑biconnected bipartite graphs (Theorem 5.1), and analogous results for critically k‑connected graphs (Theorem 6.4). These structural results enable inductive constructions that maintain maximal rank.

The paper further discusses global versions of completeness and birigidity. A graph is globally d‑completable (or globally d‑birigid) if any two generic realizations that agree on the inner products of adjacent vertices must agree on all inner products. Lemma 1.5 shows that if every vertex‑deleted subgraph is d‑completable (or d‑birigid), then the whole graph is globally d‑completable (or globally d‑birigid). Using Theorems 1.2 and 1.3 together with Lemma 1.5, Theorem 1.6 gives sufficient conditions for global d‑completeness and global d‑birigidity, which directly imply the matrix‑completion statements of Theorem 1.4.

Finally, the authors compare their results with recent work on the classical rigidity matroid R_d. Prior work by Krivelevich, Lew, and Michaeli, and by Villányi, obtained analogous degree‑ and connectivity‑based sufficient conditions for d‑rigidity. The present paper shows that similar (and in some cases stronger) conditions hold for the symmetric completion and hyperconnectivity matroids, which are closely related to R_{d‑1} through the addition of loops. The need for new extension operations and the introduction of k‑biconnectivity illustrate the distinct combinatorial challenges posed by these matroids.

In summary, the paper provides tight, degree‑based sufficient conditions for maximal rank in the symmetric completion and hyperconnectivity matroids, and a connectivity‑based sufficient condition for maximal rank in the birigidity matroid. These results translate into powerful guarantees for the unique completion of low‑rank symmetric, positive‑semidefinite, and rectangular matrices, requiring only modest amounts of observed data. The work bridges combinatorial graph theory and low‑rank matrix completion, offering both theoretical insight and practical implications for data recovery problems.