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

作者：Laurent Monique; Varvitsiotis Antonios^*

来源：Mathematical Programming, 2014, 145(1-2): 291-325.

DOI：10.1007/s10107-013-0648-x

• Semidefinite programming
• Matrix completion
• Graph realizability
• Geometric graph representation
• Euclidean distance matrix
• Graph minor
• Combinatorial matrix theory

The Gram dimension of a graph is the smallest integer such that any partial real symmetric matrix, whose entries are specified on the diagonal and at the off-diagonal positions corresponding to edges of , can be completed to a positive semidefinite matrix of rank at most (assuming a positive semidefinite completion exists). For any fixed the class of graphs satisfying is minor closed, hence it can be characterized by a finite list of forbidden minors. We show that the only minimal forbidden minor is for and that there are two minimal forbidden minors: and for . We also show some close connections to Euclidean realizations of graphs and to the graph parameter of van der Holst (Combinatorica 23(4):633-651, 2003). In particular, our characterization of the graphs with 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 of van der Holst (Combinatorica 23(4):633-651, 2003).

• 出版日期2014-6