Combinatorial Properties and Recognition of Unit Square Visibility Graphs

Katrin Casel, Henning Fernau, Alexander Grigoriev, Markus L. Schmid*, Sue Whitesides

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review


Unit square visibility graphs (USV) are described by axis-parallel visibility between unit squares placed in the plane. If the squares are required to be placed on integer grid coordinates, then USV become unit square grid visibility graphs (USGV), an alternative characterisation of the well-known rectilinear graphs. We extend known combinatorial results for USGV and we show that, in the weak case (i.e., visibilities do not necessarily translate into edges of the represented combinatorial graph), the area minimisation variant of their recognition problem is NP-hard. We also provide combinatorial insights with respect to USV, and as our main result, we prove their recognition problem to be NP-hard, which settles an open question.
Original languageEnglish
Number of pages44
JournalDiscrete & Computational Geometry
Publication statusE-pub ahead of print - Mar 2023

JEL classifications

  • c00 - Mathematical and Quantitative Methods: General


  • Geometric graph classes
  • Graph recognition
  • Visibility graphs
  • Visibility layout
  • NP-completeness

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