The excess degree of a polytope
- Authors: Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David
- Date: 2018
- Type: Text , Journal article
- Relation: SIAM Journal on Discrete Mathematics Vol. 32, no. 3 (2018), p. 2011-2046, http://purl.org/au-research/grants/arc/DP180100602
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- Description: We define the excess degree \xi (P) of a d-polytope P as 2f1 - df0, where f0 and f1 denote the number of vertices and edges, respectively. This parameter measures how much P deviates from being simple. It turns out that the excess degree of a d-polytope does not take every natural number: the smallest possible values are 0 and d - 2, and the value d - 1 only occurs when d = 3 or 5. On the other hand, for fixed d, the number of values not taken by the excess degree is finite if d is odd, and the number of even values not taken by the excess degree is finite if d is even. The excess degree is then applied in three different settings. First, it is used to show that polytopes with small excess (i.e., \xi (P) < d) have a very particular structure: provided d ot = 5, either there is a unique nonsimple vertex, or every nonsimple vertex has degree d + 1. This implies that such polytopes behave in a similar manner to simple polytopes in terms of Minkowski decomposability: they are either decomposable or pyramidal, and their duals are always indecomposable. Second, we characterize completely the decomposable d-polytopes with 2d + 1 vertices (up to combinatorial equivalence). Third, all pairs (f0, f1), for which there exists a 5-polytope with f0 vertices and f1 edges, are determined.
On the reconstruction of polytopes
- Authors: Doolittle, Joseph , Nevo, Eran , Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David
- Date: 2019
- Type: Text , Journal article
- Relation: Discrete and Computational Geometry Vol. 61, no. 2 (2019), p. 285-302. http://purl.org/au-research/grants/arc/DP180100602
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- Description: Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton. Call a vertex of a d-polytope nonsimple if the number of edges incident to it is more than d. We show that (1) the face lattice of any d-polytope with at most two nonsimple vertices is determined by its 1-skeleton; (2) the face lattice of any d-polytope with at most d- 2 nonsimple vertices is determined by its 2-skeleton; and (3) for any d> 3 there are two d-polytopes with d- 1 nonsimple vertices, isomorphic (d- 3) -skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for 4-polytopes. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.
Finite alternation theorems and a constructive approach to piecewise polynomial approximation in chebyshev norm
- Authors: Crouzeix, Jean-Pierre , Sukhorukova, Nadezda , Ugon, Julien
- Date: 2020
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 28, no. 1 (2020), p. 123-147. http://purl.org/au-research/grants/arc/DP180100602
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- Description: One of the purposes in this paper is to provide a better understanding of the alternance property which occurs in Chebyshev polynomial approximation and continuous piecewise polynomial approximation problems. In the first part of this paper, we prove that alternating sequences of any continuous function are finite in any given segment and then propose an original approach to obtain new proofs of the well known necessary and sufficient optimality conditions. There are two main advantages of this approach. First of all, the proofs are intuitive and easy to understand. Second, these proofs are constructive and therefore they lead to new alternation-based algorithms. In the second part of this paper, we develop new local optimality conditions for free knot polynomial spline approximation. The proofs for free knot approximation are relying on the techniques developed in the first part of this paper. The piecewise polynomials are required to be continuous on the approximation segment. © 2020, Springer Nature B.V.
Almost simplicial polytopes : the lower and upper bound theorems
- Authors: Nevo, Eran , Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David
- Date: 2020
- Type: Text , Journal article
- Relation: Canadian Journal of Mathematics Vol. 72, no. 2 (2020), p. 537-556. http://purl.org/au-research/grants/arc/DP180100602
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- Description: We study -vertex -dimensional polytopes with at most one nonsimplex facet with, say, vertices, called almost simplicial polytopes. We provide tight lower and upper bound theorems for these polytopes as functions of, and, thus generalizing the classical Lower Bound Theorem by Barnette and the Upper Bound Theorem by McMullen, which treat the case where s = 0. We characterize the minimizers and provide examples of maximizers for any. Our construction of maximizers is a generalization of cyclic polytopes, based on a suitable variation of the moment curve, and is of independent interest. © 2018 Canadian Mathematical Society.
Connectivity of cubical polytopes
- Authors: Bui, Hoa , Pineda-Villavicencio, Guillermo , Ugon, Julien
- Date: 2019
- Type: Text , Journal article
- Relation: Journal of Combinatorial Theory Series A Vol. 169, no. (Jan 2019), p. 21
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- Description: A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. We deal with the connectivity of the graphs of cubical polytopes. We first establish that, for any d >= 3, the graph of a cubical d-polytope with minimum degree 5 is min{delta, 2d - 2}-connected. Second, we show, for any d >= 4, that every minimum separator of cardinality at most 2d - 3 in such a graph consists of all the neighbours of some vertex and that removing the vertices of the separator from the graph leaves exactly two components, with one of them being the vertex itself. (C) 2019 Elsevier Inc. All rights reserved.
The linkedness of cubical polytopes: the cube
- Authors: Bui, Hoa , Pineda-Villavicencio, Guillermo , Ugon, Julien
- Date: 2021
- Type: Text , Journal article
- Relation: Electronic Journal of Combinatorics Vol. 28, no. 3 (2021), p.
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least 2k vertices is k-linked if, for every set of k disjoint pairs of vertices, there are k vertex-disjoint paths joining the vertices in the pairs. We say that a polytope is k-linked if its graph is k-linked. We establish that the d-dimensional cube is [(d + 1)/2]-linked, for every d ≠ 3; this is the maximum possible linkedness of a d-polytope. This result implies that, for every d ≥ 1, a cubical d-polytope is [d/2]-linked, which answers a question of Wotzlaw (Incidence graphs and unneighborly polytopes, Ph.D. thesis, 2009). Finally, we introduce the notion of strong linkedness, which is slightly stronger than that of linkedness. A graph G is strongly k-linked if it has at least 2k + 1 vertices and, for every vertex v of G, the subgraph G − v is k-linked. We show that cubical 4-polytopes are strongly 2-linked and that, for each d ≥ 1, d-dimensional cubes are strongly
Lower bound theorems for general polytopes
- Authors: Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David
- Date: 2019
- Type: Text , Journal article
- Relation: European Journal of Combinatorics Vol. 79, no. (2019), p. 27-45
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: For a d-dimensional polytope with v vertices, d + 1 <= v <= 2d, we calculate precisely the minimum possible number of m-dimensional faces, when m = 1 or m >= 0.62d. This confirms a conjecture of Grunbaum, for these values of m. For v = 2d + 1, we solve the same problem when m = 1 or d - 2; the solution was already known for m = d - 1. In all these cases, we give a characterisation of the minimising polytopes. We also show that there are many gaps in the possible number of m-faces: for example, there is no polytope with 80 edges in dimension 10, and a polytope with 407 edges can have dimension at most 23.
Polytopes close to being simple
- Authors: Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David
- Date: 2020
- Type: Text , Journal article
- Relation: Discrete and Computational Geometry Vol. 64, no. 1 (2020), p. 200-215
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: It is known that polytopes with at most two nonsimple vertices are reconstructible from their graphs, and that d-polytopes with at most d- 2 nonsimple vertices are reconstructible from their 2-skeletons. Here we close the gap between 2 and d- 2 , showing that certain polytopes with more than two nonsimple vertices are reconstructible from their graphs. In particular, we prove that reconstructibility from graphs also holds for d-polytopes with d+ k vertices and at most d- k+ 3 nonsimple vertices, provided k