Embeddings of k-Complexes into 2k-Manifolds
Autoři
Paták, P.; Tancer, M.
Rok
2024
Publikováno
Discrete & Computational Geometry. 2024, 71(3), 960-991. ISSN 0179-5376.
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In this paper we identify a necessary algebraic condition for a simplicial k-complex to be embeddable into a 2k-manifold M. For k>=3 and sufficiently connected M, this condition is also sufficient. We further use this condition to improve known bounds for Kühnel's problem: given a 2k-manifold M, what is the largest n such that the k-skeleton of n-simplex can be embedded into M? Based on these results we also show variants of Helly's and Radon's theorems for certain families of sets in M.
Jordan-Hölder Theorem with Uniqueness for Semimodular Lattices
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In 2011 Czédli and Schmidt proved the strongest form of Jordan-Hölder theorem for lattices, which they called Jordan-Hölder theorem with uniqueness: Given two maximal chains in a semimodular lattice of finite height, they both have the same length and there is a unique bijection that takes the prime intervals of the first chain to the prime intervals of the second chain such that the interval and its image are up-and-down projective. The theorem generalizes the classical result that all composition series of a finite group have the same length and isomorphic factors and shows that the isomorphism is in some sense unique. The paper presents a simplified proof of the result.
Shellability Is Hard Even for Balls
Autoři
Paták, P.; Tancer, M.
Rok
2023
Publikováno
STOC 2023: Proceedings of the 55th Annual ACM Symposium on Theory of Computing. New York: Association for Computing Machinery, 2023. p. 1271-1284. ISBN 978-1-4503-9913-5.
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The main goal of this paper is to show that shellability is NP-hard for triangulated d-balls (this also gives hardness for triangulated d-manifolds/d-pseudomanifolds with boundary) as soon as d ≥ 3. This extends our earlier work with Goaoc, Patáková and Wagner on hardness of shellability of 2-complexes and answers some questions implicitly raised by Danaraj and Klee in 1978 and explicitly mentioned by Santamaría-Galvis and Woodroofe. Together with the main goal, we also prove that collapsibility is NP-hard for 3-complexes embeddable in 3-space, extending an earlier work of the second author and answering an open question mentioned by Cohen, Fasy, Miller, Nayyeri, Peng and Walkington; and that shellability is NP-hard for 2-complexes embeddable in 3-space, answering another question of Santamaría-Galvis and Woodroofe (in a slightly stronger form than what is given by the main result).
Disjoint Compatibility via Graph Classes
Autoři
Aichholzer, O.; Obmann, J.; Paták, P.; Perz, D.; Tkadlec, J.; Vogtenhuber, B.
Rok
2022
Publikováno
Graph-Theoretic Concepts in Computer Science. Springer, Cham, 2022. p. 16-28. Lecture Notes in Computer Science. vol. 13453. ISSN 0302-9743. ISBN 978-3-031-15913-8.
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Two plane drawings of graphs on the same set of points are called disjoint compatible if their union is plane and they do not have an edge in common Let S be a convex point set of 2n >= 10 points and let H be a family of plane drawings on S. Two plane perfect matchings M-1 and M-2 on S (which do not need to be disjoint nor compatible) are disjoint H-compatible if there exists a drawing in H which is disjoint compatible to both M-1 and M-2. In this work, we consider the graph which has all plane perfect matchings as vertices and where two vertices are connected by an edge if the matchings are disjoint H-compatible. We study the diameter of this graph when H is the family of all plane spanning trees, caterpillars or paths. We show that in the first two cases the graph is connected with constant and linear diameter, respectively, while in the third case it is disconnected.