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 Kn admits a 2.5D-BR if and only if n≤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−6n 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 Γ is the set of
bottom faces of the boxes in Γ.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016