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NONPROJECTABILITY OF POLYTOPE SKELETA
, 2009
"... We investigate necessary conditions for the existence of projections of polytopes that preserve full kskeleta. More precisely, given the combinatorics of a polytope and the dimension e of the target space, what are obstructions to the existence of a geometric realization of a polytope with the giv ..."
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Cited by 4 (1 self)
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We investigate necessary conditions for the existence of projections of polytopes that preserve full kskeleta. More precisely, given the combinatorics of a polytope and the dimension e of the target space, what are obstructions to the existence of a geometric realization of a polytope with the given combinatorial type such that a linear projection to espace strictly preserves the kskeleton. Building on the work of Sanyal (2009), we develop a general framework to calculate obstructions to the existence of such realizations using topological combinatorics. Our obstructions take the form of graph colorings and linear integer programs. We focus on polytopes of product type and calculate the obstructions for products of polygons, products of simplices, and wedge products of polytopes. Our results show the limitations of constructions for the deformed products of polygons of Sanyal & Ziegler (2009) and the wedge product surfaces of Rörig & Ziegler (2009) and complement their results.
Polytopality and Cartesian products of graphs
 Accepted in Israel Journal of Mathematics
"... Abstract. We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of po ..."
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Cited by 1 (1 self)
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Abstract. We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of nonpolytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (nonsimple) polytopal products whose factors are not polytopal. Even though graphs are perhaps the most prominent feature of polytopes, we are still far from being able to answer several basic questions regarding them. For applications, one of the most important ones is to bound the diameter of the graph in terms of the number of variables and inequalities defining the polytope [San10]. From a theoretical point of view, it is striking that we cannot even efficiently decide whether a given graph occurs as the graph of a polytope or not [RG96].
PRODSIMPLICIALNEIGHBORLY POLYTOPES
, 908
"... Abstract. We introduce PSN polytopes whose kskeleton is combinatorially equivalent to that of a product of r simplices. They simultaneously generalize both neighborly and neighborly cubical polytopes. We construct PSN polytopes by three different methods, the most versatile of which is an extension ..."
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Abstract. We introduce PSN polytopes whose kskeleton is combinatorially equivalent to that of a product of r simplices. They simultaneously generalize both neighborly and neighborly cubical polytopes. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal & Ziegler’s “projecting deformed products ” construction to products of arbitrary simple polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1. Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that this dimension is minimal if we moreover require the PSN polytope to be obtained as a projection of a polytope combinatorially equivalent to the product of r simplices, when the sum of their dimensions is at least 2k. 1.
DOI: 10.1007/s004540109311y PRODSIMPLICIALNEIGHBORLY POLYTOPES
, 2013
"... Abstract. Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their kskeleton is combinatorially equivalent to that of a product of r simplices. We construct PSN polytopes by three different methods, the most versatile of which is an extension o ..."
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Abstract. Simultaneously generalizing both neighborly and neighborly cubical polytopes, we introduce PSN polytopes: their kskeleton is combinatorially equivalent to that of a product of r simplices. We construct PSN polytopes by three different methods, the most versatile of which is an extension of Sanyal & Ziegler’s “projecting deformed products ” construction to products of arbitrary simple polytopes. For general r and k, the lowest dimension we achieve is 2k+r+1. Using topological obstructions similar to those introduced by Sanyal to bound the number of vertices of Minkowski sums, we show that this dimension is minimal if we additionally require that the PSN polytope is obtained as a projection of a polytope that is combinatorially equivalent to the product of r simplices, when the dimensions of these simplices are all large compared to k. 1.
DOI: 10.1007/s1185601200495 POLYTOPALITY AND CARTESIAN PRODUCTS OF GRAPHS
, 2013
"... Abstract. We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of po ..."
Abstract
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Abstract. We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of nonpolytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (nonsimple) polytopal products whose factors are not polytopal. Even though graphs are perhaps the most prominent feature of polytopes, we are still far from being able to answer several basic questions regarding them. For applications, one of the most important ones is to bound the diameter of the graph in terms of the number of variables and inequalities defining the polytope [San10]. From a theoretical point of view, it is striking that we cannot even efficiently decide whether a given graph occurs as the graph of a polytope or not [RG96].