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Given a set of n points in the plane, any β-skeleton and [γ0,γ1]-graph can be computed in quadratic time. The presented algorithms are optimal for β values that are less than 1 and [γ0,γ1] values that result in non-planar graphs. We show a numerically robust algorithm that computes Gabriel graphs in quadratic time and degree 2. We finally show how a β-spectrum can be computed in optimal O(n2) time. [Copyright &y& Elsevier]

Visibility representations are a well-known paradigm to represent graphs. From a high-level perspective, a visibility representation of a graph G maps the vertices of G to non-overlapping geometric objects and the edges of G to visibilities, i.e., segments that do not intersect any geometric object other than at their end-points. In this paper, we initiate the study of 2.5D box visibility representations (2.5D-BRs) 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; ( i i ) The complete graph K n admits a 2.5D-BR if and only if n ⩽ 19 ; ( i i i ) 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-GBRs), 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 4 n − 6 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 Γ is the set of bottom faces of the boxes in Γ). [ABSTRACT FROM AUTHOR]