Decompositions of Simplicial Complexes Related to Diameters of Convex Polyhedra

TLDR: It is shown that several cases in which the Hirsch conjecture has been verified can be handled by these methods, which also give the shellability of a number of simplicial complexes of combinatorial interest.

Abstract: We introduce the property of k-decomposability for simplicial complexes which, if satisfied by the dual simplicial complex of a convex polyhedron, implies that the diameter of the polyhedron is bounded by a polynomial of degree k + 1 in the number of facets. These properties form a hierarchy, each one implying the next. The strongest, vertex decomposability, implies the Hirsch conjecture; the weakest is equivalent to shellability. We show that several cases in which the Hirsch conjecture has been verified can be handled by these methods, which also give the shellability of a number of simplicial complexes of combinatorial interest. We conclude with a strengthened form of the property of shellability which would imply the Hirsch conjecture for polytopes.