Realized Rank Certificates for Matchstick Frameworks and Insertion Edges

Abstract: We give an exact, checkable rank-certificate method for realized planar unit-distance frameworks. The method is motivated by Vogel's computations for matchstick graphs and by the insertion-edge tests used in the Matchstick Graphs Calculator. Its algebraic core is independent of geometry. A singular square matrix $M$ is replaced by a sparse perturbation $B=M+UCV^T$. If $B$ is nonsingular and the inverse satisfies $V^TB^{-1}U=C^{-1}$, then the columns of $B^{-1}U$ and the rows of $V^TB^{-1}$ form bases of the right and left kernels of $M$, and the rank defect of $M$ is certified. Applied to the equilibrium matrix of a planar framework, this gives finite exact certificates for self-stresses, infinitesimal motions, redundant edges, and candidate edges whose constraints are already forced by the realized framework. The certificate data can be checked independently from the search that produced it, using exact matrix identities. We emphasize that matchstick frameworks are not generic: unit distances, triangles, rhombi, and symmetries can change the realized rank. The method therefore concerns the coordinate-dependent representation of a given drawing, not only the generic rigidity matroid of the abstract graph.