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We give a signed generalization of Laurent's theorem that characterizesfeasible positive semidefinite matrix completion problems in terms of metricpolytopes. Based on this result, we give a characterization of the maximum rankcompletions of the signed positive semidefinite matrix completion problem forodd-$K_4$ minor free signed graphs. The analysis can also be used to bound theminimum rank over the completions and to characterize uniquely solvablecompletion problems for odd-$K_4$ minor free signed graphs. As a corollary wederive a characterization of the universal rigidity of odd-$K_4$ minor freespherical tensegrities, and also a characterization of signed graphs whosesigned Colin de Verdi\`ere parameter $\nu$ is bounded by two, recently shown byArav et al.

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