eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-22
8:1
8:18
10.4230/LIPIcs.SWAT.2022.8
article
Recognizing Map Graphs of Bounded Treewidth
Angelini, Patrizio
1
https://orcid.org/0000-0002-7602-1524
Bekos, Michael A.
2
https://orcid.org/0000-0002-3414-7444
Da Lozzo, Giordano
3
https://orcid.org/0000-0003-2396-5174
Gronemann, Martin
4
https://orcid.org/0000-0003-2565-090X
Montecchiani, Fabrizio
5
https://orcid.org/0000-0002-0543-8912
Tappini, Alessandra
5
https://orcid.org/0000-0001-9192-2067
Department of Mathematics, Natural, and Applied Sciences, John Cabot University, Rome, Italy
Department of Mathematics, University of Ioannina, Greece
Department of Engineering, Roma Tre University, Rome, Italy
Algorithms and Complexity Group, Technische Universität Wien, Austria
Department of Engineering, University of Perugia, Italy
A map graph is one admitting a representation in which vertices are nations on a spherical map and edges are shared curve segments or points between nations. We present an explicit fixed-parameter tractable algorithm for recognizing map graphs parameterized by treewidth. The algorithm has time complexity that is linear in the size of the graph and, if the input is a yes-instance, it reports a certificate in the form of a so-called witness. Furthermore, this result is developed within a more general algorithmic framework that allows to test, for any k, if the input graph admits a k-map (where at most k nations meet at a common point) or a hole-free k-map (where each point is covered by at least one nation). We point out that, although bounding the treewidth of the input graph also bounds the size of its largest clique, the latter alone does not seem to be a strong enough structural limitation to obtain an efficient time complexity. In fact, while the largest clique in a k-map graph is ⌊ 3k/2 ⌋, the recognition of k-map graphs is still open for any fixed k ≥ 5.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol227-swat2022/LIPIcs.SWAT.2022.8/LIPIcs.SWAT.2022.8.pdf
Map graphs
Recognition
Parameterized complexity