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 t...

In this paper we introduce the quasi-upward planar drawing convention and give a polynomial time algorithm for computing a quasiupward planar drawing with the minimum number of bends within a given planar embedding. Further, we study the problem of computing quasi-upward planar drawings with the minimum number of bends of digraphs considering all the possible planar embeddings. The paper contai...

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...

We analyze a directed variation of the book embedding problem when the page partition is prespecified and the nodes on the spine must be in topological order (upward book embedding). Given a directed acyclic graph and a partition of its edges into k pages, can we linearly order the vertices such that the drawing is upward (a topological sort) and each page avoids crossings? We prove that the pr...

Given an embedded planar acyclic digraph G, we define the problem of acyclic hamiltonian path completion with crossing minimization (Acyclic-HPCCM) to be the problem of determining a hamiltonian path completion set of edges such that, when these edges are embedded on G, they create the smallest possible number of edge crossings and turn G to an acyclic hamiltonian digraph. Our results include: ...

Upward planar drawings of digraphs are crossing free drawings where all edges flow in the upward direction. The problem of deciding whether a digraph admits an upward planar drawing is called the upward planarity testing problem, and it has been widely studied in the literature. In this paper we investigate a new upward planarity testing problem, that is, deciding whether a digraph admits an up...

An upward topological book embedding of a planar st-digraph G is an upward planar drawing of G such that its vertices are aligned along the vertical line, called the spine, and each edge is represented as a simple Jordan curve which is divided by the intersections with the spine (spine crossings) into segments such that any two consecutive segments are located at opposite sides of the spine. Wh...

Given an embedded planar acyclic digraph G, we define the problem of"acyclic hamiltonian path completion with crossing minimization (Acyclic-HPCCM)"to be the problem of determining an hamiltonian path completion set of edges such that, when these edges are embedded on G, they create the smallest possible number of edge crossings and turn G to a hamiltonian digraph. Our results include: --We pro...

We study level planarity testing of graphs with a fixed combinatorial embedding for three different notions embeddings, namely the embedding, upward and planar embedding. These allow increasing degrees freedom in their corresponding drawings. For there are known easy to test criteria. use these criteria prove an untangling lemma that plays key role simple case where only is fixed. This then ada...