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arXiv · 1603.08370

The Signed Positive Semidefinite Matrix Completion Problem for Odd-$K_4$ Minor Free Signed Graphs

Shin-ichi Tanigawa
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Abstract

We give a signed generalization of Laurent's theorem that characterizes feasible positive semidefinite matrix completion problems in terms of metric polytopes. Based on this result, we give a characterization of the maximum rank completions of the signed positive semidefinite matrix completion problem for odd-$K_4$ minor free signed graphs. The analysis can also be used to bound the minimum rank over the completions and to characterize uniquely solvable completion problems for odd-$K_4$ minor free signed graphs. As a corollary we derive a characterization of the universal rigidity of odd-$K_4$ minor free spherical tensegrities, and also a characterization of signed graphs whose signed Colin de Verdi\`ere parameter $\nu$ is bounded by two, recently shown by Arav et al.

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BibTeXRIS

Shin-ichi Tanigawa. 2016-03-28. The Signed Positive Semidefinite Matrix Completion Problem for Odd-$K_4$ Minor Free Signed Graphs. https://arxiv.org/abs/1603.08370

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