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Abstract: This paper investigates the basic problem of computing crossing-free straight-line 3D grid drawings of graphs such that the overall volume is small. Motivated by their relevance in the literature, we focus on families of graphs having constant queue number and on k-trees. We present algorithms that compute drawings of these families of graphs on 3D grids consisting of a constant number of parallel lines and such that the overall volume is linear. Lower bounds on the number of such grid lines are also provided. Our results extend and improve similar ones already described in the literature. [Copyright &y& Elsevier]

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]

Abstract: This paper studies the problem of computing orthogonal drawings of graphs with labels on vertices and edges. Our research is mainly motivated by Software Engineering and Information Systems domains, where tools like UML diagrams and ER-diagrams are considered fundamental for the design of sophisticated systems and/or complex data bases collecting enormous amount of information. A label is modeled as a rectangle of prescribed width and height and it can be associated with either a vertex or an edge. Our drawing algorithms guarantee no overlaps between labels, vertices, and edges and take advantage of the information about the set of labels to compute the geometry of the drawing. Several additional optimization goals are taken into account. Namely, the labeled drawing can be required to have either minimum total edge length, or minimum width, or minimum height, or minimum area among those preserving a given orthogonal representation. All these goals lead to NP-hard problems. We present MILP models to compute optimal drawings with respect to the first three goals and an exact algorithm that is based on these models to compute a labeled drawing of minimum area. We also present several heuristics for computing compact labeled orthogonal drawings and experimentally validate their performances, comparing their solutions against the optimum. [Copyright &y& Elsevier]

Let C be the family of 2D curves described by concave functions, let G be a planar graph, and let L be a linear ordering of the vertices of G. L is a curve embedding of G if for any given curve Λ∈C there exists a planar drawing of G such that: (i) the vertices are constrained to be on Λ with the same ordering as in L, and (ii) the edges are polylines with at most one bend. Informally speaking, a curve embedding can be regarded as a two-page book embedding in which the spine is bent. Although deciding whether a graph has a two-page book embedding is an NP-hard problem, in this paper it is proven that every planar graph has a curve embedding which can be computed in linear time. Applications of the concept of curve embedding to upward drawability and point-set embeddability problems are also presented. [Copyright &y& Elsevier]