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 ≤ 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~(1/2) 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.
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