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On the Complexity of Recognizing k^+-Real Face Graphs

Authors Michael A. Bekos , Giuseppe Di Battista , Emilio Di Giacomo , Walter Didimo , Michael Kaufmann , Fabrizio Montecchiani

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ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Graph theory
  • Theory of computation → Graph algorithms analysis
Keywords
  • Beyond planarity
  • k^+-real face drawings
  • 2-layer drawings
  • recognition algorithm
  • NP-hardness

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Abstract

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.

Cite As Get BibTex

Michael A. Bekos, Giuseppe Di Battista, Emilio Di Giacomo, Walter Didimo, Michael Kaufmann, and Fabrizio Montecchiani. On the Complexity of Recognizing k^+-Real Face Graphs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 32:1-32:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.32

Author Details

Michael A. Bekos
  • University of Ioannina, Greece
Giuseppe Di Battista
  • University of Roma Tre, Italy
Emilio Di Giacomo
  • University of Perugia, Italy
Walter Didimo
  • University of Perugia, Italy
Michael Kaufmann
  • University of Tübingen, Germany
Fabrizio Montecchiani
  • University of Perugia, Italy

Funding

  • Di Battista, Giuseppe: MUR PRIN Proj. 2022TS4Y3N - EXPAND: scalable algorithms for EXPloratory Analyses of heterogeneous and dynamic Networked Data.
  • Didimo, Walter: AIDMIX - Artificial Intelligence for Decision Making: Methods for Interpretability and eXplainability - Ricerca di Base 2021. RASTA: Realtà Aumentata e Story-Telling Automatizzato per la valorizzazione di Beni Culturali ed Itinerari; Italian MUR PON Proj. ARS01_00540.
  • Montecchiani, Fabrizio: MUR PRIN Proj. 2022ME9Z78 - NextGRAAL: Next-generation algorithms for constrained GRAph visuALization.

Acknowledgements

Research started at the Dagstuhl Seminar: Beyond-Planar Graphs: Models, Structures and Geometric Representations; seminar number: 24062, February 4-9, 2024.

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