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

Monique Laurent, Antonios Varvitsiotis
2012 arXiv   pre-print
The Gram dimension (G) of a graph G is the smallest integer k> 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 (G) < k is minor closed, hence it can characterized by a finite list of forbidden minors. We show that the only
more » ... inimal forbidden minor is K_k+1 for k< 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 ν^=(G) of H03. In particular, our characterization of the graphs with (G)< 4 implies the forbidden minor characterization of the 3-realizable graphs of Belk and Connelly Belk,BC and of the graphs with ν^=(G) < 4 of van der Holst H03.
arXiv:1204.0734v1 fatcat:y7eue5ijkrejzg2j7dz5uy7aue