A nonplanar drawing Γ of a graph G divides the plane into topologically connected regions, called faces (or cells). The boundary of each face is formed by vertices, crossings, and edge segments. Given a positive integer k, we say that Γ is a k^+-real face drawing of G if the boundary of each face of Γ contains at least k vertices of G. The study of k^+-real face drawings started in a paper by Binucci et al. (WG 2023), where edge density bounds and relationships with other beyond-planar graph classes are proved. In this paper, we investigate the complexity of recognizing k^+-real face graphs, i.e., graphs that admit a k^+-real face drawing. We study both the general unconstrained scenario and the 2-layer scenario in which the graph is bipartite, the vertices of the two partition sets lie on two distinct horizontal layers, and the edges are straight-line segments. We give NP-completeness results for the unconstrained scenario and efficient recognition algorithms for the 2-layer setting.
@InProceedings{bekos_et_al:LIPIcs.GD.2024.32, author = {Bekos, Michael A. and Di Battista, Giuseppe and Di Giacomo, Emilio and Didimo, Walter and Kaufmann, Michael and Montecchiani, Fabrizio}, title = {{On the Complexity of Recognizing k^+-Real Face Graphs}}, booktitle = {32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)}, pages = {32:1--32:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-343-0}, ISSN = {1868-8969}, year = {2024}, volume = {320}, editor = {Felsner, Stefan and Klein, Karsten}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.32}, URN = {urn:nbn:de:0030-drops-213167}, doi = {10.4230/LIPIcs.GD.2024.32}, annote = {Keywords: Beyond planarity, k^+-real face drawings, 2-layer drawings, recognition algorithm, NP-hardness} }
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