Balinski's theorem

In polyhedral combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of three-dimensional convex polyhedra and higher-dimensional convex polytopes. It states that, if one forms an undirected graph from the vertices and edges of a convex d-dimensional convex polyhedron or polytope (its skeleton), then the resulting graph is at least d-vertex-connected: the removal of any d − 1 vertices leaves a connected subgraph. For instance, for a three-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.^[1]

Balinski's theorem is named after mathematician Michel Balinski, who published its proof in 1961,^[2] 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.^[3]

Balinski proves the result based on the correctness of the simplex method for finding the minimum or maximum of a linear function on a convex polytope (the linear programming problem). The simplex method starts at an arbitrary vertex of the polytope and repeatedly moves towards an adjacent vertex that improves the function value; when no improvement can be made, the optimal function value has been reached.

If S is a set of fewer than d vertices to be removed from the graph of the polytope, Balinski adds one more vertex v₀ to S and finds a linear function ƒ that has the value zero on the augmented set but is not identically zero on the whole space. Then, any remaining vertex at which ƒ is non-negative (including v₀) can be connected by simplex steps to the vertex with the maximum value of ƒ, while any remaining vertex at which ƒ is non-positive (again including v₀) can be similarly connected to the vertex with the minimum value of ƒ. Therefore, the entire remaining graph is connected.