A new graph parameter related to bounded rank positive semidefinite matrix completions

The Gram dimension $\gd(G)$ of a graph $G$ is the smallest integer $k\ge 1$ such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the off-diagonal positions corresponding to edges of $G$, can be completed to a positive semidefinite matrix of rank at most $k$ (assuming a positive semidefinite completion exists). For any fixed $k$ the class of graphs satisfying $\gd(G) \le k$ is minor closed, hence it can characterized by a finite list of forbidden minors. We show that the only minimal forbidden minor is $K_{k+1}$ for $k\le 3$ and that there are two minimal forbidden minors: $K_5$ and $K_{2,2,2}$ for $k=4$. We also show some close connections to Euclidean realizations of graphs and to the graph parameter $\nu^=(G)$ of \cite{H03}. In particular, our characterization of the graphs with $\gd(G)\le 4$ implies the forbidden minor characterization of the 3-realizable graphs of Belk and Connelly \cite{Belk,BC} and of the graphs with $\nu^=(G) \le 4$ of van der Holst \cite{H03}.

💡 Research Summary

The paper introduces a new graph invariant called the Gram dimension, denoted gd(G). For a graph G, gd(G) is the smallest integer k such that any partially specified real symmetric matrix—where entries are known on the diagonal and on off‑diagonal positions corresponding to edges of G—admits a positive semidefinite (PSD) completion of rank at most k, provided a PSD completion exists at all. This parameter captures the minimal rank needed to realize the given partial inner‑product information, and therefore directly relates to Euclidean realizations of graphs.

A fundamental structural property proved is that the class of graphs with gd(G) ≤ k is minor‑closed. Consequently, by the Robertson‑Seymour theory, each such class can be characterized by a finite set of forbidden minors. The authors determine these forbidden minors completely for the first non‑trivial cases. For k ≤ 3 they show that the only minimal forbidden minor is the complete graph K_{k+1}. The proof proceeds by computing gd(K_{k+1}) = k+1 and then demonstrating that any graph not containing K_{k+1} as a minor can be completed to a PSD matrix of rank ≤ k, using tree‑width arguments and rank‑reduction techniques.

When k = 4 the situation becomes richer. The paper proves that exactly two graphs are minimal forbidden minors: the complete graph K_5 and the complete tripartite graph K_{2,2,2}. For K_5 the argument follows the same line as for smaller k. For K_{2,2,2} the authors exploit the known fact that this graph is not 3‑realizable in Euclidean space (a result of Belk and Connelly). They show that any PSD completion respecting the edge pattern of K_{2,2,2} necessarily has rank at least 5, establishing gd(K_{2,2,2}) = 5.

The work also uncovers tight connections with two previously studied parameters. First, it proves that a graph is 3‑realizable (i.e., can be embedded in ℝ³ with prescribed edge lengths) if and only if gd(G) ≤ 4, thereby providing a new forbidden‑minor characterization of the 3‑realizable graphs originally obtained by Belk and Connelly. Second, it shows that gd(G) ≤ 4 is equivalent to ν^{=}(G) ≤ 4, where ν^{=}(G) is the parameter introduced by van der Holst in the context of the Colin de Verti‑type invariants. This equivalence links the Gram dimension to spectral graph theory and to the theory of graph minors.

In summary, the paper delivers a complete forbidden‑minor description for graphs of Gram dimension at most four, establishes that for k ≤ 3 the only obstruction is K_{k+1}, and identifies K_5 and K_{2,2,2} as the two obstructions for k = 4. These results unify concepts from semidefinite programming, Euclidean graph realizations, and minor theory, and they set the stage for future investigations into higher values of k, algorithmic computation of gd(G), and potential applications in rigidity theory and low‑rank matrix completion.