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Many papers study the natural problem of drawing nonplanar graphs with few crossings per edge. In particular, a graph is 1-planar if it can be drawn in the plane with at most one crossing per edge. Unfortunately, while testing graph planarity is solvable in linear time and several efficient algorithms have been described in the literature, deciding whether a graph is 1-planar is NP-complete, even for restricted classes of graphs. Despite some polynomial-time algorithms are known for recognizing specific subfamilies of 1-planar graphs, there is still a lack of practical 1-planarity testing algorithms and no implementation is available for general graphs. This paper investigates the feasibility of a 1-planarity testing and embedding algorithm based on a backtracking strategy. Our contribution provides initial indications that have the potential to stimulate further research on the design of practical approaches for the 1-planarity testing problem. On the one hand, our experiments show that a backtracking strategy can be successfully applied to graphs with up to 30 vertices. On the other hand, our study suggests that alternative techniques are needed to attack larger graphs. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study how to compute compact straight-line drawings of planar graphs with a limited number of crossings per edge. We prove that every outerplanar graph can be drawn in area using a sub-linear number of crossings per edge, and that for any given number , every outerplanar graph admits an area drawing with crossing per edge. The drawing algorithms run in linear time and can be also generalized to another meaningful sub-family of series–parallel graphs with bounded vertex-degree. [Copyright &y& Elsevier]

Abstract: Given a graph G with n vertices and a set S of n points in the plane, a point-set embedding of G on S is a planar drawing such that each vertex of G is mapped to a distinct point of S. A geometric point-set embedding is a point-set embedding with no edge bends. This paper studies the following problem: The input is a set S of n points, a planar graph G with n vertices, and a geometric point-set embedding of a subgraph on a subset of S. The desired output is a point-set embedding of G on S that includes the given partial drawing of . We concentrate on trees and show how to compute the output in time in a real-RAM model and with at most edges with at most bends, where k is the number of vertices of the given subdrawing. We also prove that there are instances of the problem which require at least bends on edges. [Copyright &y& Elsevier]

Abstract: Let G be an embedded planar digraph. A maximum upward planar subgraph of G is an embedding preserving subgraph that is upward planar and, among those, has the maximum number of edges. This paper presents an extensive study on the problem of computing maximum upward planar subgraphs of embedded planar digraphs: Complexity results, algorithms, and experiments are presented. Namely: (i) we prove that the addressed problem is NP-Hard; (ii) a fast heuristic and an exponential-time exact algorithm are described; (iii) a wide experimental analysis is performed to show the effectiveness of our techniques. [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]

Abstract: In this paper we study the problem of computing an upward straight-line embedding of a planar DAG (directed acyclic graph) G into a point set S, i.e. a planar drawing of G such that each vertex is mapped to a point of S, each edge is drawn as a straight-line segment, and all the edges are oriented according to a common direction. In particular, we show that no biconnected DAG admits an upward straight-line embedding into every point set in convex position. We provide a characterization of the family of DAGs that admit an upward straight-line embedding into every convex point set such that the points with the largest and the smallest y-coordinate are consecutive in the convex hull of the point set. We characterize the family of DAGs that contain a Hamiltonian directed path and that admit an upward straight-line embedding into every point set in general position. We also prove that a DAG whose underlying graph is a tree does not always have an upward straight-line embedding into a point set in convex position and we describe how to construct such an embedding for a DAG whose underlying graph is a path. Finally, we give results about the embeddability of some sub-classes of DAGs whose underlying graphs are trees on point set in convex and in general position. [Copyright &y& Elsevier]

We initiate the study of the following problem: Given a non-planar graph G and a planar subgraph S of G, does there exist a straight-line drawing Γ of G in the plane such that the edges of S are not crossed in Γ by any edge of G? We give positive and negative results for different kinds of connected spanning subgraphs S of G . Moreover, in order to enlarge the subset of instances that admit a solution, we consider the possibility of bending the edges of G not in S ; in this setting we discuss different trade-offs between the number of bends and the required drawing area. [ABSTRACT FROM AUTHOR]