Structural characterization of arbitrary matrix-completion stresses

Characterize arbitrary matrix-completion stresses, including for $K_4$-minor-free graphs, by identifying a structural description comparable to the known decomposition for positive semidefinite stresses.

Background

The paper interprets degeneracy of a completion through nonzero matrix-completion stresses. For positive semidefinite completions of cycle-completable graphs, the authors note that the cone of positive semidefinite stresses is generated by stresses supported on maximal cliques and chordless cycles. They show that this structural description fails for unrestricted complex or real stresses, already for $K_{2,3}$, where a nonzero stress exists that cannot be expressed using stresses supported on maximal cliques or induced cycles.

The absence of a structural characterization creates a difficulty in proving the paper’s main theorem: the argument must derive a clique or cycle degeneracy condition directly from the existence of an arbitrary nonzero stress. The problem therefore asks for a general structural theory of such stresses, even within the restricted class of $K_4$-minor-free graphs.