Kalai, G.: A simple way to tell a simple polytope from its graph, J. Combin. Theory Ser. A 49(2) (1988), 381--383.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the (complete) graphs obtained by shrinking all (n 1) cliques in G # and H # (which, again, does not produce multiple edges) This, finally, yields an isomorphism between G and H . The equivalence of (ii) and (iv) follows from a theorem of Blind and Mani [4] see also Kalai s beautiful proof [17]) stating that two simple polytopes are isomorphic if and only if their graphs are isomorphic. For the special polytopes arising from our construction, the equivalence can, however, be alternatively deduced similarly to the proof of (ii) i) Statements (iv) and (v) obviously are ....

Kalai, G.: A simple way to tell a simple polytope from its graph. J. Comb. Theory Ser. A 49, 381--383 (1988)

....GP corresponding to a nonempty face of P . A US orientation is called an abstract objective function orientation (AOF orientation) if it is acyclic. General US orientations of graphs of cubes have recently received some attention (Szab o and Welzl [61] AOForientations were used, e.g. by Kalai [35]. Since their linear extensions are precisely the shelling orders of the dual polytope, they have been considered much earlier. 13. Face Lattice of Combinatorial Polytopes Solvable in O(minfm;ng ) time, where m is the number of facets, n is the number of vertices, is the number of ....

....Mani [6] proved that the entire combinatorial structure of a simple polytope is determined by its graph. This is false for general polytopes (of dimension at least four) which is the main reason why we restrict our attention to simple polytopes for the remaining problems in this section. Kalai [35] gave a short, elegant, and constructive proof of Blind and Mani s result. However, the algorithm that can be derived from it has a worst case running time that is exponential in the number of vertices of the polytope. In [32] it is shown that the problem can be formulated as a combinatorial ....

G. Kalai, A simple way to tell a simple polytope from its graph, J. Comb. Theory, Ser. A, 49 (1988), pp. 381-383.

....the (complete) graphs obtained by shrinking all (n 1) cliques in G # and H # (which, again, does not produce multiple edges) This, finally, yields an isomorphism between G and H . The equivalence of (ii) and (iv) follows from a theorem of Blind and Mani [4] see also Kalai s beautiful proof [17]) stating that two simple polytopes are isomorphic if and only if their graphs are isomorphic. For the special polytopes arising from our construction, the equivalence can, however, be alternatively deduced similarly to the proof of (ii) i) Statements (iv) and (v) obviously are ....

Kalai, G.: A simple way to tell a simple polytope from its graph. J. Comb. Theory Ser. A 49, 381--383 (1988)

....10623 Berlin, Germany kaibel math.tu berlin.de http: www.math.tu berlin.de kaibel Abstract. Blind and Mani [2] proved that the entire combinatorial structure (the vertex facet incidences) of a simple convex polytope is determined by its abstract graph. Their proof is not constructive. Kalai [14] found a short, elegant, and algorithmic proof of that result. However, his algorithm has an exponential running time. We show that the problem to reconstruct the vertex facet incidences of a simple polytope P from its graph can be formulated as a combinatorial optimization problem that is ....

....of a polytope from its graph, is simple. In fact, Blind and Mani [2] proved in 1987 that the face lattice of a simple polytope is determined by its graph. Their proof (which we sketch in Sect. 2) is not constructive and crucially relies on the topological concept of homology. In 1988, Kalai [14] found a short and elegant proof (reviewed in Sect. 3) that does only use elementary geometric and combinatorial reasoning with the main advantage of being algorithmic. However, the running time of the method that can be devised from it is exponential in the size of the graph. Perles conjectured ....

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G. Kalai. A simple way to tell a simple polytope from its graph. J. Comb. Theory, Ser. A, 49(2):381-383, 1988.

....certi cate MSC 2000: 52B11 (n dimensional polytopes) 52B05 (combinatorial properties) 1 Introduction A celebrated theorem of Blind and Mani [2] states that the combinatorial type of any simple polytope P is determined by the isomorphism class of its abstract vertex edge graph GP . Kalai [8] gave a short and very elegant proof of this result. The proof is constructive, but the algorithm that can be derived from it has a worst case running time which is exponential in the size of GP (for computational experiments see Achatz and Kleinschmidt [1] Thus, the complexity status of the ....

....d X k=0 H k (O) The sum H k (O) is the number of k frames for which all edges are directed towards the root. Thus H k (O) is the total number of sinks induced in the subgraphs GP (S) of GP (S 2 S) One of the beautiful steps on Kalai s Simple Way to Tell a Simple Polytope from its Graph [8] (see also [10] Chap. 3.4) is the observation that the AOF orientations of GP are precisely those orientations that minimize H (O) Theorem 5 implies that AOF orientations can also be characterized as those acyclic orientations of GP that minimize H 2 (O) If O is an AOF orientation of GP , then ....

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G. Kalai, A simple way to tell a simple polytope from its graph, J. Comb. Theory, Ser. A 49 (1988), 381-383.

....again, does not produce multiple edges) This, nally, yields an isomorphism between G and G 0 . 5 The equivalence of (ii) and (iv) follows immediately from the proof of the rst equivalence, but can also be obtained from a theorem of Blind and Mani [3] see also Kalai s beautiful proof [11]) stating that two simple polytopes are isomorphic if and only if their graphs are isomorphic. Statements (iv) and (v) obviously are equivalent. Unlike the situation for simple polytopes, it is, in general, not true that two (simplicial) polytopes are isomorphic if and only their graphs are ....

G. Kalai, A simple way to tell a simple polytope from its graph, J. Comb. Theory, Ser. A., 49 (1988), pp. 381-383.

....faces. Even when there exist a full description, see [1] it is often general combinatorial, and useless for fast numerical computations. On the other hand, it is well known that if the polytope is simple, then theoretically one can obtain the number of k dimensional faces from its graph (see [7,15]) Again, known algorithms run in the exponential time (cf. 10] In this paper we present an efficient algorithm for computing the number of faces of transportation polytopes in a special case when n = m 1, a = m 1; m 1) and b = m; m) Denote this polytope Pm . The dimension ....

G. Kalai, A simple way to tell a simple polytope from its graph, J. Comb. Th., Ser A 49 (1988), 381--383.

....faces. Even when there exists a full description (see [1] it is often general combinatorial, and useless for fast numerical computations. On the other hand, it is well known that if the polytope is simple, then theoretically one can obtain the number of k dimensional faces from its graph (see [7, 15]) Again, known algorithms run in exponential time (cf. 10] In this paper we present an efficient algorithm for computing the number of faces of transportation polytopes in a special case when n = m 1, a = m 1, m 1) and b = m, m) Denote this polytope Pm . ....

G. Kalai, A simple way to tell a simple polytope from its graph, J. Comb. Theory, Ser. A, 49 (1988), 381--383.

....and Convex Sets. 1 Introduction Let M be an n pseudomanifold with boundary. In the dual graph of M denoted G(M ) vertices correspond to the n cells of M with an edge between two vertices iff the corresponding n cells share an (n Gamma 1) cell. Micha A. Perles asked the following question [1] : Let C be a subset of facets of a simplicial d polytope P , and C the complement of C. If both G(C) and G( C) are connected and if G(C) is (d Gamma 1) regular then must C necessarily be the star of a vertex We ask the same question in the more general setting of triangulated spheres (instead ....

Kalai G, A Simple Way to Tell a Simple Polytope from its Graph, Jour. Comb. Theory Series A 49 (1988) 237-239.

....subclass can be recovered from its graph, by applying our results recursively. 1 Introduction Let P be a simple d polytope and G(P ) the graph (1 skeleton) of P . Perles conjectured that every (d Gamma 1) regular, induced, connected and non separating subgraph of G(P ) determines a facet of P [2]. In this paper we prove the conjecture for a proper subclass of simple polytopes. The motivation for our results comes from two subclasses of simplicial polytopes, namely the stacked polytopes and the crosspolytopes. Polytopes obtained from a simplex by successive addition of pyramids over ....

G. Kalai. A simple way to tell a simple polytope from its graph; Jour. Comb. Theory, Series A 49, 381-383, 1988.

....faces. Even when there exist a full description, see [1] it is often general combinatorial, and useless for fast numerical computations. On the other hand, it is well known that if the polytope is simple, then theoretically one can obtain the number of k dimensional faces from its graph (see [7,15]) Again, known algorithms run in the exponential time (cf. 10] In this paper we present an ecient algorithm for computing the number of faces of transportation polytopes in a special case when n = m 1, a = m 1; m 1) and b = m; m) Denote this polytope Pm . The dimension of ....

G. Kalai, A simple way to tell a simple polytope from its graph, J. Comb. Th., Ser A 49 (1988), 381-383.

....rank 3 case, our main result implies as well a theorem of Goodman and Pollack, namely [8, Theorem 2. 9] It should be mentioned that our results can be considered as variations of Blind Mani s theorem [4] that says the one skeleton of a simple convex polytope determines its face lattice (see also [11]) When an oriented matroid is representable, the complex W coincides with the face lattice of the dual of a (d 1) zonotope. Thus our results show how theorems of Blind Mani s type can be phrased for a certain class of non simple polytopes. For a long time our results remain unpublished, ....

Kalai, Gil: "A simple way to tell a simple polytope from its graph", J. Combin. Theory Ser. A 49 (1988), no. 2, 381--383.

....polytope, f vector, g vector, labeled trees, computing. Typeset by A M S T E X 2 I. PAK and useless for fast numerical computations. On the other hand, it is well known that if the polytope is simple, then theoretically one can obtain the number of k dimensional faces from its graph (see [Ka,Z]) Again, known algorithms run in the exponential time (cf. Kr] In this paper we present an efficient algorithm for computing the number of faces of transportation polytopes in a special case when n = m 1, a = m 1; m 1) and b = m; m) Denote this polytope Pm . The dimension ....

G. Kalai, A simple way to tell a simple polytope from its graph, J. Comb. Th., Ser A 49 (1988), 381--383.

....puzzle based on the local information: which two simplices share a facet. Joswig extended their result to more genaral puzzles where the pieces are general (d 1) dimensional polytopes and the way every two pieces which share a facet are connected is also prescribed. A simple proof is given in [54]. This proofs also shows that k dimensional skeletons of simplicial polytopes are also determined by their puzzle . Combined with Perles theorem it follows that: THEOREM 19.5.25 Kalai and Perles (1988) Simplicial d polytopes are determined by the incidence relations between i and (i ....

G. Kalai. A simple way to tell a simple polytope from its graph. J. Combinatorial Theory, Ser. A, 49:381-383, 1988.

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G. Kalai, A simple way to tell a simple polytope from its graph, J. Comb. Th. (Ser. A) 49(1988), 381-383.

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G. Kalai, A simple way to tell a simple polytope from its graph, J. Comb. Th. (Ser. A) 49(1988), 381-383.

....determines the face structure, and Dancis [14] extended this result to arbitrary simplicial spheres. Perles conjectured and Blind and Mani [11] proved that the face structure of every simple d polytope is determined by the graph (1 skeleton) of the polytope. For a simple proof see, Kalai [26]. Consider a simplicial (d Gamma 1) dimensional sphere and a puzzle in which the pieces are the facets and for each piece there is a list of the d neighboring pieces. The Blind Mani theorem asserts that for boundary complexes of simplicial polytopes (and for a certain class of shellable spheres) ....

G. Kalai, A simple way to tell a simple polytope from its graph, J. Comb. Th. (Ser. A) 49(1988), 381-383.

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Kalai, G.: A simple way to tell a simple polytope from its graph, J. Combin. Theory Ser. A 49(2) (1988), 381--383.

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G. Kalai. A simple way to tell a simple polytope from its graph. J. Combin. Theory Ser. A, 49:381-383, 1988.

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G. Kalai, A simple way to tell a simple polytope from its graph, J. Combinatorial Theory, Ser. A, 49 (1988), pp. 381--383.

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