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• W2024448984 abstract "The Gram dimension $$mathrm{gd}(G)$$ of a graph $$G$$ is the smallest integer $$kge 1$$ 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 $$kle 3$$ 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)." @default.
• W2024448984 created "2016-06-24" @default.
• W2024448984 creator A5047819350 @default.
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• W2024448984 date "2013-02-16" @default.
• W2024448984 modified "2023-09-25" @default.
• W2024448984 title "A new graph parameter related to bounded rank positive semidefinite matrix completions" @default.
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• W2024448984 doi "https://doi.org/10.1007/s10107-013-0648-x" @default.
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