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

TL;DR

This paper introduces a new graph parameter called Gram dimension, characterizes its minor-closed classes for small values, and explores its connections to Euclidean graph realizations and other graph parameters.

Contribution

It provides a complete forbidden minor characterization of graphs with Gram dimension at most 4, extending understanding of positive semidefinite matrix completions.

Findings

Minimal forbidden minors for $ ext{gd}(G) extless= 3$ are $K_{k+1}$.

Forbidden minors for $ ext{gd}(G) extless= 4$ are $K_5$ and $K_{2,2,2}$.

Connections established between Gram dimension, Euclidean realizations, and the parameter $ u^=(G)$.

Abstract

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}.