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Combinatorial Properties and Recognition of Unit Square Visibility Graphs.
Katrin Casel1, Henning Fernau2, Alexander Grigoriev3
1Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany.
Discrete & Computational Geometry
|May 14, 2023
Summary
This study investigates visibility graphs of unit squares. The recognition problem for unit square visibility graphs is proven to be NP-hard, settling an open question in computational geometry.
Area of Science:
- Computational Geometry
- Graph Theory
- Combinatorics
Background:
- Unit square visibility graphs (USV) model axis-parallel visibility between unit squares.
- Unit square grid visibility graphs (USGV) are a specific type of USV on integer grids, related to rectilinear graphs.
Purpose of the Study:
- To extend combinatorial results for USGV.
- To determine the computational complexity of recognizing USV and USGV.
- To settle an open question regarding the NP-hardness of USV recognition.
Main Methods:
- Extending combinatorial results for USGV.
- Analyzing the weak case of USGV recognition for area minimization.
- Proving the NP-hardness of the USV recognition problem.
Main Results:
- The area minimization variant of USGV recognition is NP-hard (weak case).
- The recognition problem for general unit square visibility graphs (USV) is NP-hard.
- New combinatorial insights for USV are provided.
Conclusions:
- The recognition of unit square visibility graphs is computationally complex (NP-hard).
- The findings contribute to understanding the complexity of geometric visibility problems.
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