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VCDimensions For Graphs
, 1994
"... We study set systems over the vertex set (or edge set) of some graph that are induced by special graph properties like clique, connectedness, path, star, tree, etc. We derive a variety of combinatorial and computational results on the VC (VapnikChervonenkis) dimension of these set systems. For most ..."
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Cited by 1 (0 self)
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to these intractability results, we show that the VCdimension for set systems induced by stars is computable in polynomial time. For set systems induced by paths or cycles, we determine the extremal graphs G with the minimum number of edges such that VCP (G) k. Finally, we show a close relation between the VC
The VCDimension of Graphs with Respect to kConnected Subgraphs
"... We study the VCdimension of the set system on the vertex set of some graph which is induced by the family of its kconnected subgraphs. In particular, we give upper and lower bounds for the VCdimension. Moreover, we show that computing the VCdimension is NPcomplete and that it remains NPcomplet ..."
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We study the VCdimension of the set system on the vertex set of some graph which is induced by the family of its kconnected subgraphs. In particular, we give upper and lower bounds for the VCdimension. Moreover, we show that computing the VCdimension is NPcomplete and that it remains NP
Asymptotic estimates for best and stepwise approximation of convex bodies III
, 1997
"... . We consider approximations of a smooth convex body by inscribed and circumscribed convex polytopes as the number of vertices, resp. facets tends to infinity. The measure of deviation used is the difference of the mean width of the convex body and the approximating polytopes. The following results ..."
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Cited by 58 (4 self)
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are obtained. (i) An asymptotic formula for best approximation. (ii) Upper and lower estimates for stepby step approximation in terms of the socalled dispersion. (iii) For a sequence of best approximating inscribed polytopes the sequence of vertex sets is uniformly distributed in the boundary of the convex
Representing polynomials by positive linear functions on compact convex polyhedra
 Pacific Journal of Mathematics
, 1988
"... If K is a compact polyhedron in Euclidean έ/space, defined by linear inequalities, βt> 0, and if / is a polynomial in d variables that is strictly positive on AT, then / can be expressed as a positive linear combination of products of members of {/?,}. In proving this and subsidiary results, we ..."
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Cited by 43 (0 self)
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If K is a compact polyhedron in Euclidean έ/space, defined by linear inequalities, βt> 0, and if / is a polynomial in d variables that is strictly positive on AT, then / can be expressed as a positive linear combination of products of members of {/?,}. In proving this and subsidiary results, we
Linkages in Polytope Graphs
, 2007
"... A graph is klinked if any k disjoint vertexpairs can be joined by k disjoint paths. We improve a lower bound on the linkedness of polytopes slightly, which results in exact values for the minimal linkedness of 7, 10 and 13dimensional polytopes. We analyze in detail linkedness of polytopes on at ..."
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A graph is klinked if any k disjoint vertexpairs can be joined by k disjoint paths. We improve a lower bound on the linkedness of polytopes slightly, which results in exact values for the minimal linkedness of 7, 10 and 13dimensional polytopes. We analyze in detail linkedness of polytopes
ON THE GRAPHCONNECTIVITY OF SKELETA OF CONVEX POLYTOPES
, 2008
"... Given a ddimensional convex polytope P and nonnegative integer k not exceeding d − 1, let Gk(P) denote the simple graph on the node set of kdimensional 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 isom ..."
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Cited by 3 (2 self)
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(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 mvertexconnected for all ddimensional polytopes P is determined. This result generalizes Balinski’s theorem on the onedimensional skeleton of a d
Polytopes in Arrangements
, 1999
"... Consider an arrangement of n hyperplanes in R d . Families of convex polytopes whose boundaries are contained in the union of the hyperplanes are the subject of this paper. We aim to bound their combinatorial complexity. Exact asymptotic bounds were known for the case where the polytopes are cells ..."
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Cited by 2 (1 self)
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vertices. In R 3 we show an O(k 1=3 n 2 ) bound on the number of faces of k such polytopes. We also discuss worstcase lower bounds and higherdimensional versions of the problem. Among other results, we show that the maximum number of facets of k pairwise vertexdisjoint polytopes in R d is k 1
Practical Methods for Approximating Shortest Paths on a Convex Polytope in R³
 COMPUT. GEOM. THEORY APPL
, 1995
"... We propose an extremely simple approximation scheme for computing shortest paths on the surface of a convex polytope in three dimensions. Given a convex polytope P with n vertices and two points p; q on its surface, let dP (p; q) denote the shortest path distance between p and q on the surface of P ..."
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Cited by 19 (1 self)
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We propose an extremely simple approximation scheme for computing shortest paths on the surface of a convex polytope in three dimensions. Given a convex polytope P with n vertices and two points p; q on its surface, let dP (p; q) denote the shortest path distance between p and q on the surface
The Graph of the Hypersimplex
, 2008
"... The (k, d)hypersimplex is a (d − 1)dimensional polytope whose vertices are the (0, 1)vectors that sum to k. When k = 1, we get a simplex whose graph is the complete graph Kd. Here we show how many of the well known graph parameters and attributes of Kd extend to a more general case. In particular ..."
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The (k, d)hypersimplex is a (d − 1)dimensional polytope whose vertices are the (0, 1)vectors that sum to k. When k = 1, we get a simplex whose graph is the complete graph Kd. Here we show how many of the well known graph parameters and attributes of Kd extend to a more general case. In particular
CONNECTIVITY OF PSEUDOMANIFOLD GRAPHS FROM AN ALGEBRAIC POINT OF VIEW
"... Abstract. The connectivity of graphs of simplicial and polytopal complexes is a classical subject going back at least to Steinitz, and the topic has since been studied by many authors, including Balinski, Barnette, Athanasiadis and Björner. In this note, we provide a unifying approach which allows u ..."
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that the underlying graph of a dpolytope is dconnected, cf. [Zie95]. David Barnette showed that the same bound is also valid for the connectivity number of underlying graphs of (d − 1)dimensional pseudomanifolds [Bar82]. Athanasiadis [Ath11] showed that if the pseudomanifold is also flag (i.e. the clique complex
Results 1  10
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27