Balinski's theorem [SEP] named after
dimensional polyhedron, even if two of its vertices (together with their incident edges) are removed, for any pair of vertices there will still exist a path of vertices and edges connecting the pair.
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.
Balinski's proof
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Balinski's Ph.D. thesis concerned the vertex enumeration problem, the algorithmic problem of listing all vertices of a convex polytope or finding all optimal solutions of a linear program, and some of his subsequent work continued to concern polyhedral combinatorics. The thesis includes the fundamental theorem, published in 1961, that the skeletons of polytopes in "n"-space viewed as graphs are "n"-connected, meaning that at least "n" edges must be removed to disconnect the graph of the remaining vertices and edges; it is known as Balinski's theorem