Visibility Representations of Boxes in 2.5 Dimensions

Alessio Arleo; Carla Binucci; Emilio Di Giacomo; William S. Evans; Luca Grilli; Giuseppe Liotta; Henk Meijer; Fabrizio Montecchiani; Sue Whitesides; Stephen Wismath

arXiv · 1608.08899

Visibility Representations of Boxes in 2.5 Dimensions

Alessio Arleo·Carla Binucci·Emilio Di Giacomo·William S. Evans·Luca Grilli·Giuseppe Liotta·Henk Meijer·Fabrizio Montecchiani·Sue Whitesides·Stephen Wismath

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Abstract

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 $n \leq 19$; $(iii)$ 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 $\Gamma$ is the set of bottom faces of the boxes in $\Gamma$.

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Alessio Arleo, Carla Binucci, Emilio Di Giacomo, William S. Evans, Luca Grilli, Giuseppe Liotta, Henk Meijer, Fabrizio Montecchiani, Sue Whitesides, Stephen Wismath. 2016-08-31. Visibility Representations of Boxes in 2.5 Dimensions. https://arxiv.org/abs/1608.08899

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