Shellability Is Hard Even for Balls
Authors
Paták, P.; Tancer, M.
Year
2023
Published
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.
Type
Proceedings paper
Departments
Annotation
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
Authors
Aichholzer, O.; Obmann, J.; Paták, P.; Perz, D.; Tkadlec, J.; Vogtenhuber, B.
Year
2022
Published
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.
Type
Proceedings paper
Departments
Annotation
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.