Balinski's theorem

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In polyhedral combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of three-dimensional polyhedra and higher-dimensional polytopes. It states that, if one forms an undirected graph from the vertices and edges of a convex d-dimensional 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's proof[edit]

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.

References[edit]

  1. ^ Ziegler, Günter M. (1995), "Section 3.5: Balinski's Theorem: The Graph is d-Connected", Lectures on Polytopes, Graduate Texts in Mathematics 152, Springer-Verlag .
  2. ^ Balinski, M. L. (1961), "On the graph structure of convex polyhedra in n-space", Pacific Journal of Mathematics 11 (2): 431–434, doi:10.2140/pjm.1961.11.431, MR 0126765 .
  3. ^ Steinitz, E. (1922), "Polyeder und Raumeinteilungen", Encyclopädie der mathematischen Wissenschaften, Band 3 (Geometries), pp. 1–139 .