Visibility Representations of Boxes in 2.5 Dimensions

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 $\Gamma$.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

Area Requirement of Visibility Representations of Trees

Abstract We study the area requirement of bar-visibility and rectangle-visibility representations of trees in the plane. We prove asymptotically tight lower and upper bounds on the area of such representations, and give linear-time algorithms that construct representations with asymptotically optimal area. 1 Introduction Visibility is a fundamental relation in computational geometry (see, e.g., [1, 8, 9]). In particular, the problem of constructing visibility representations of graphs, where the vertices are drawn as geometric objects of a certain class and the edges are associated with pairs of "visible " objects, has been extensively investigated (see, e.g., [3, 5, 12])