3D global rigidity characterization via 3-thin, 4-shellable covers (excluding K_{5,5})
Show that for any graph G on at least five vertices that is not K_{5,5}, G is globally rigid in R^3 if and only if for every edge set F ⊆ E(G) and every 3-thin, 4-shellable cover X of E(G) \ F with vertex-sets of size ≥ 5, the inequality |F| + Σ_{X∈X}(3|X| − 6) − Σ_{h∈H(X)}(d_X(h) − 1) ≥ 3|V(G)| − 6 holds, with equality only when F = ∅ and X = {V(G)}.
References
Conjecture\nLet $G=(V,E)$ be a graph on at least five vertices which is not a copy of $K_{5,5}$. Then $G$ is globally rigid in $3$ if and only if, for all $F\subseteq E$ and all\n3-thin, 4-shellable covers of $E \setminus F$, $\mathcal{X}$, $|F|+\, ()\geq 3|V|-6$\nwith equality only when $F=\emptyset$ and $={V}$.
— Rigidity of Graphs and Frameworks: A Matroid Theoretic Approach
(2508.11636 - Cruickshank et al., 29 Jul 2025) in Section 4.3 (Global Rigidity)