Balinski's Theorem describes the graph-theoretic structure of convex polyhedra and higher-dimensional convex polytopes, stating that the graph formed from the vertices and edges of a d-dimensional convex polyhedron or polytope is at least d-vertex-connected, so removing any d minus one vertices leaves the remaining graph connected. Named after Michel Balinski, who published its proof in 1961, the three-dimensional case of the result traces back earlier in the century to Steinitz's theorem characterizing the graphs of three-dimensional polyhedra as the three-connected planar graphs.
Facts
StatementThe skeleton graph formed from the vertices and edges of any d dimensional convex polyhedron or polytope is at least d vertex connected, so removing any fewer than d vertices leaves the remaining graph connected. 1 Classification
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Balinski's theorem (Wikipedia)
Sources
1. Balinski's theorem (Wikipedia)
Wikimedia FoundationLead section
Balinski's theorem is named after mathematician Michel Balinski, who published its proof in 1961, although the three-dimensional case dates back to the earlier part of the 20th century and the discovery of Steinitz's theorem that the graphs of three-dimensional polyhedra are exactly the three-connected planar graphs.
In Branch: Combinatorics, Lead sentence
In polyhedral combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of thr
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