Theorem of the valine ski

A theorem (valine ski のていり British: Balinski's theorem) of the valine ski in the polyhedron combinatorics (English version) that is a field for one minute of the mathematics is a theorem about the structure of the graph theory to have of three-dimensional polyhedron and more high-dimensional ポリトープ. When I form a graph for nothing from a top and the side of a certain d- dimension convex polyhedron or ポリトープ (skeleton (English version)), it is the theorem that spoke that the graph is at least d- top connection (i.e., the left graph is connected even if I remove the top of any d-1 unit). For example, a top and the paths of the side which tie them to the pair of the left arbitrary top exist even if two (the side that and connects with them) of the tops were removed for the polyhedron with three dimensions [¹].

The theorem of the valine ski is associated with the name of the Michel L valine ski of the mathematician who gave the proof in 1961 [²]. However, as for the graph of the three-dimensional polyhedron, a result was provided as a theorem (English version) of シュタイニッツ to be a 3-connection plane graph about the three-dimensional case in the early 20th century [³].

Proof of the valine ski

The proof of the valine ski depends on the accuracy of the simplex method on finding the minimum of the linear function on convex ポリトープ or maximum (linear programming problem). By the simplex method, I leave the top with the option included in ポリトープ and repeat the movement to the adjacent top improving the value of the function. It looks like it; when was not able to be improved, it means that a most suitable function level is provided.

When I did S with the set consisting of tops of numbers less than d which should be removed from the graph of ポリトープ, I add another top v₀ to it by the proof of the valine ski, and it is 0, but a value finds alignment function ƒ which is not 0 identically in all space on the meeting. Then a value of ƒ virtually becomes the non-plus on the arbitrary top where there remained it, and, in the top (including v₀) had the option that ƒ becomes the non-minus number for, in (after all including v₀), a value of ƒ is connected to the minimized top likewise while a value of ƒ is connected to the top becoming greatest by the simplex method. As a result, the graph left is a connection altogether.

References

1. ^ Ziegler, Günter M. (1995), "Section 3.5 is Lectures on Polytopes, Graduate Texts in Mathematics, 152, Springer-Verlag Balinski's Theorem: The Graph is d-Connected".
2. ^ Balinski, M. L. (1961), "On the graph structure of convex polyhedra in n-space," it is Pacific Journal of Mathematics 11 (2): 431–434, MR 0126765, http://projecteuclid.org/euclid.pjm/1103037323 .
3. ^ Steinitz, E. (1922), "Polyeder und Raumeinteilungen," it is Encyclopädie der mathematischen Wissenschaften, Band 3 (Geometries), pp. 1–139.

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