Visibility Representations of Boxes in 2.5 Dimensions

  • Alessio Arleo
  • Carla Binucci
  • Emilio Di GiacomoEmail author
  • William S. Evans
  • Luca Grilli
  • Giuseppe Liotta
  • Henk Meijer
  • Fabrizio Montecchiani
  • Sue Whitesides
  • Stephen Wismath
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9801)


We initiate the study of 2.5D box visibility representations (2.5D-BR) where vertices are mapped to 3D boxes having the bottom face in the plane \(z=0\) and edges are unobstructed lines of sight parallel to the x- or y-axis. We prove that: (i) Every complete bipartite graph admits a 2.5D-BR; (ii) The complete graph \(K_n\) admits a 2.5D-BR if and only if \(n \leqslant 19\); (iii) Every graph with pathwidth at most 7 admits a 2.5D-BR, which can be computed in linear time. We then turn our attention to 2.5D grid box representations (2.5D-GBR) which are 2.5D-BRs such that the bottom face of every box is a unit square at integer coordinates. We show that an n-vertex graph that admits a 2.5D-GBR has at most \(4n - 6 \sqrt{n}\) edges and this bound is tight. Finally, we prove that deciding whether a given graph G admits a 2.5D-GBR with a given footprint is NP-complete. The footprint of a 2.5D-BR \(\varGamma \) is the set of bottom faces of the boxes in \(\varGamma \).


Visibility Representation Complete Graph Geometric Object Hamiltonian Path Complete Bipartite Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Alessio Arleo
    • 1
  • Carla Binucci
    • 1
  • Emilio Di Giacomo
    • 1
    Email author
  • William S. Evans
    • 2
  • Luca Grilli
    • 1
  • Giuseppe Liotta
    • 1
  • Henk Meijer
    • 3
  • Fabrizio Montecchiani
    • 1
  • Sue Whitesides
    • 4
  • Stephen Wismath
    • 5
  1. 1.Università degli Studi di PerugiaPerugiaItaly
  2. 2.University of British ColumbiaVancouverCanada
  3. 3.University College RooseveltMiddelburgThe Netherlands
  4. 4.University of VictoriaVictoriaCanada
  5. 5.University of LethbridgeLethbridgeCanada

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