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

Abstract

The Gram dimension $$\mathrm{gd}(G)$$ 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 $$\mathrm{gd}(G) \le k$$ is minor closed, hence it can be 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 van der Holst (Combinatorica 23(4):633–651, 2003). In particular, our characterization of the graphs with $$\mathrm{gd}(G)\le 4$$ implies the forbidden minor characterization of the 3-realizable graphs of Belk (Discret Comput Geom 37:139–162, 2007) and Belk and Connelly (Discret Comput Geom 37:125–137, 2007) and of the graphs with $$\nu ^=(G) \le 4$$ of van der Holst (Combinatorica 23(4):633–651, 2003).