5 research outputs found
Hierarchical Partial Planarity
In this paper we consider graphs whose edges are associated with a degree of
{\em importance}, which may depend on the type of connections they represent or
on how recently they appeared in the scene, in a streaming setting. The goal is
to construct layouts of these graphs in which the readability of an edge is
proportional to its importance, that is, more important edges have fewer
crossings. We formalize this problem and study the case in which there exist
three different degrees of importance. We give a polynomial-time testing
algorithm when the graph induced by the two most important sets of edges is
biconnected. We also discuss interesting relationships with other
constrained-planarity problems.Comment: Conference version appeared in WG201
On Optimal 2- and 3-Planar Graphs
A graph is -planar if it can be drawn in the plane such that no edge is
crossed more than times. While for , optimal -planar graphs, i.e.,
those with vertices and exactly edges, have been completely
characterized, this has not been the case for . For and ,
upper bounds on the edge density have been developed for the case of simple
graphs by Pach and T\'oth, Pach et al. and Ackerman, which have been used to
improve the well-known "Crossing Lemma". Recently, we proved that these bounds
also apply to non-simple - and -planar graphs without homotopic parallel
edges and self-loops.
In this paper, we completely characterize optimal - and -planar graphs,
i.e., those that achieve the aforementioned upper bounds. We prove that they
have a remarkably simple regular structure, although they might be non-simple.
The new characterization allows us to develop notable insights concerning new
inclusion relationships with other graph classes
Colored anchored visibility representations in 2D and 3D space
© 2020. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/In a visibility representation of a graph G, the vertices are represented by nonoverlapping geometric objects, while the edges are represented as segments that only intersect the geometric objects associated with their end-vertices. Given a set P of n points, an Anchored Visibility Representation of a graph G with n vertices is a visibility representation such that for each vertex v of G, the geometric object representing v contains a point of P. We prove positive and negative results about the existence of anchored visibility representations under various models, both in 2D and in 3D space. We consider the case when the mapping between the vertices and the points is not given and the case when it is only partially given.Peer ReviewedPostprint (author's final draft
Planar and Quasi Planar Simultaneous Geometric Embedding
A simultaneous geometric embedding (SGE) of two planar graphs G 1 and G 2 with the same vertex set is a pair of straight-line planar drawings Î1 of G 1 and Î2 of G 2 such that each vertex is drawn at the same point in Î1 and Î2. Many papers have been devoted to the study of which pairs of graphs admit a SGE, and both positive and negative results have been proved. We extend the study of SGE, by introducing and characterizing a new class of planar graphs that makes it possible to immediately extend several positive results that rely on the property of strictly monotone paths. Moreover, we introduce a relaxation of the SGE setting where Î1 and Î2 are required to be quasi planar (i.e., they can have crossings provided that there are no three mutually crossing edges). This relaxation allows for the simultaneous embedding of pairs of planar graphs that are not simultaneously embeddable in the classical SGE setting and opens up to several new interesting research questions