Computing the number of realisations of a rigid graph
Sean Dewar, Georg Grasegger, Josef Schicho, Ayush Kumar Tewari, Audie Warren
TL;DR
The paper solves the problem of computing the realisation number $c_{2}(G)$ for any rigid planar graph by introducing a recursive, combinatorial formula that expresses $c_{2}(G)$ in terms of counts on smaller graphs. Leveraging an algebraic-geometry framework, it proves that $c_{2}(G)$ equals the degree of the graph map $p_G$ and that this degree factors according to edge-decomposition and 2-cut structures, enabling a complete computation by recursion. It further extends the framework to spherical realisations with $c_{2}^{\circ}(G)$ and relates rank-3 PSD matrix completions to complex completions, providing analogous counting algorithms for $r=3$. The work is supplemented by computational results up to graphs with ten vertices, derivations of growth bounds via generalized fan constructions, and an openly available dataset and code, highlighting both theoretical and practical impact for rigidity theory and matrix completion tasks.
Abstract
A graph is said to be rigid if, given a generic realisation of the graph as a bar-and-joint framework in the plane, there exist only finitely many other realisations of the graph with the same edge lengths modulo rotations, reflections and translations. In recent years there has been an increase of interest in determining exactly what this finite amount is, hereon known as the realisation number. Combinatorial algorithms for the realisation number were previously known for the special cases of minimally rigid and redundantly rigid graphs. In this paper we provide a combinatorial algorithm to compute the realisation number of any rigid graph, and thus solve an open problem of Jackson and Owen. We then adapt our algorithm to compute: (i) spherical realisation numbers, and (ii) the number of rank-3 PSD matrix completions of a generic partial matrix.