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Summary: Discrete Comput Geom (2009) 42: 155165
DOI 10.1007/s00454-009-9181-3
On the Graph Connectivity of Skeleta of Convex
Polytopes
Christos A. Athanasiadis
Received: 23 June 2008 / Revised: 27 February 2009 / Accepted: 8 April 2009 /
Published online: 12 May 2009
© Springer Science+Business Media, LLC 2009
Abstract Given a d-dimensional convex polytope P and nonnegative integer k not
exceeding d - 1, let Gk(P) denote the simple graph on the node set of k-dimensional
faces of P in which two such faces are adjacent if there exists a (k + 1)-dimensional
face of P which contains them both. The graph Gk(P) is isomorphic to the dual graph
of the (d - k)-dimensional skeleton of the normal fan of P. For fixed values of k and
d, the largest integer m such that Gk(P) is m-vertex-connected for all d-dimensional
polytopes P is determined. This result generalizes Balinski's theorem on the one-
dimensional skeleton of a d-dimensional convex polytope.
Keywords Convex polytope · Balinski's theorem · Incidence graph · Skeleton ·
Vertex connectivity · Cell complex
1 Introduction
The combinatorial theory of convex polytopes has provided mathematicians with in-
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