Planarity-preserving clustering and embedding for large planar graphs (original) (raw)
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Drawing Planar Graphs with Large Vertices and Thick Edges
Journal of Graph Algorithms and Applications, 2004
We consider the problem of representing size information in the edges and vertices of a planar graph. Such information can be used, for example, to depict a network of computers and information traveling through the network. We present an efficient linear-time algorithm which draws edges and vertices of varying 2-dimensional areas to represent the amount of information flowing through them. The algorithm avoids all occlusions of nodes and edges, while still drawing the graph on a compact integer grid.
Simultaneous embedding of planar graphs
Simultaneous embedding is concerned with simultaneously representing a series of graphs sharing some or all vertices. This forms the basis for the visualization of dynamic graphs and thus is an important field of research. Recently there has been a great deal of work investigating simultaneous embedding problems both from a theoretical and a practical point of view. We survey recent work on this topic. * Submitted as a chapter about simultaneous embedding to the GD Handbook edited by Roberto Tamassia.
Recognizing and Drawing IC-Planar Graphs
Lecture Notes in Computer Science, 2015
We give new results about the relationship between 1-planar graphs and RAC graphs. A graph is 1-planar if it has a drawing where each edge is crossed at most once. A graph is RAC if it can be drawn in such a way that its edges cross only at right angles. These two classes of graphs and their relationships have been widely investigated in the last years, due to their relevance in application domains where computing readable graph layouts is important to analyze or design relational data sets. We study ICplanar graphs, the sub-family of 1-planar graphs that admit 1-planar drawings with independent crossings (i.e., no two crossed edges share an endpoint). We prove that every IC-planar graph admits a straight-line RAC drawing, which may require however exponential area. If we do not require right angle crossings, we can draw every ICplanar graph with straight-line edges in linear time and quadratic area. We then study the problem of testing whether a graph is IC-planar. We prove that this problem is NPhard, even if a rotation system for the graph is fixed. On the positive side, we describe a polynomial-time algorithm that tests whether a triangulated plane graph augmented with a given set of edges that form a matching is IC-planar.
An algorithm for straight-line drawing of planar graphs
1998
Abstract. We present a new algorithm for drawing planar graphs on the plane. It can be viewed as a generalization of the algorithm of Chrobak and Payne, which, in turn, is based on an algorithm by de Fraysseix, Pach, and Pollack. Our algorithm improves the previous ones in that it does not require a preliminary triangulation step; triangulation proves problematic in drawing graphs``nicely,''as it has the tendency to ruin the structure of the input graph.
Windrose planarity: embedding graphs with direction-constrained edges
Symposium on Discrete Algorithms, 2016
Given a planar graph G(V, E) and a partition of the neighbors of each vertex v ∈ V in four sets v, v, v, and v, the problem WINDROSE PLANARITY asks to decide whether G admits a windrose-planar drawing, that is, a planar drawing in which (i) each neighbor u ∈ v is above and to the right of v, (ii) each neighbor u ∈ v is above and to the left of v, (iii) each neighbor u ∈ v is below and to the left of v, (iv) each neighbor u ∈ v is below and to the right of v, and (v) edges are represented by curves that are monotone with respect to each axis. By exploiting both the horizontal and the vertical relationship among vertices, windrose-planar drawings allow to simultaneously visualize two partial orders defined by means of the edges of the graph. Although the problem is N P-hard in the general case, we give a polynomial-time algorithm for testing whether there exists a windrose-planar drawing that respects a combinatorial embedding that is given as part of the input. This algorithm is based on a characterization of the plane triangulations admitting a windrose-planar drawing. Furthermore, for any embedded graph admitting a windrose-planar drawing we show how to construct one with at most one bend per edge on an O(n) × O(n) grid. The latter result contrasts with the fact that straight-line windrose-planar drawings may require exponential area.
Convex drawings of hierarchical planar graphs and clustered planar graphs
Journal of Discrete Algorithms, 2010
Hierarchical graphs are graphs with layering structures; clustered graphs are graphs with recursive clustering structures. Both have applications in VLSI design, CASE tools, software visualisation and visualisation of social networks and biological networks. Straight-line drawing algorithms for hierarchical graphs and clustered graphs have been presented in [P. Eades, Q. Feng, X. Lin and H. Nagamochi, Straight-line drawing algorithms for hierarchical graphs and clustered graphs, Algorithmica, 44, pp. 1-32, 2006].
Arxiv preprint arXiv: …, 2012
In a drawing of a clustered graph vertices and edges are drawn as points and curves, while clusters are represented by simple closed regions. A drawing is c-planar if it has no edge-edge, edge-region, or region-region crossings. An obvious necessary condition for c-planarity is the planarity of the graph underlying the clustered graph. However, planarity is not sufficient and the constraints imposed by the absence of edge-region and of region-region crossings make the family of c-planar graphs too small for some of the typical Graph Drawing application contexts. Hence, we relax such constraints and define and study α, β, γdrawings of c-graphs whose underlying graph is planar. In an α, β, γdrawing the number of edge-edge, edge-region, and region-region crossings is equal to α, β, and γ, respectively. In this context α, β, γ -drawings are a generalization of c-planar drawings, where α = β = γ = 0.
Box-Rectangular Drawings of Planar Graphs
Journal of Graph Algorithms and Applications, 2013
A plane graph is a planar graph with a fixed planar embedding in the plane. In a box-rectangular drawing of a plane graph, every vertex is drawn as a rectangle, called a box, each edge is drawn as either a horizontal line segment or a vertical line segment, and the contour of each face is drawn as a rectangle. A planar graph is said to have a box-rectangular drawing if at least one of its plane embeddings has a box-rectangular drawing. Rahman et al. [11] gave a necessary and sufficient condition for a plane graph to have a box-rectangular drawing and developed a lineartime algorithm to draw a box-rectangular drawing of a plane graph if it exists. Since a planar graph G may have an exponential number of planar embeddings, determining whether G has a box-rectangular drawing or not using the algorithm of Rahman et al. [11] for each planar embedding of G takes exponential time. Thus to develop an efficient algorithm to examine whether a planar graph has a box-rectangular drawing or not is a non-trivial problem. In this paper we give a linear-time algorithm to determine whether a planar graph G has a box-rectangular drawing or not, and to find a box-rectangular drawing of G if it exists.
Matched Drawings of Planar Graphs
Journal of Graph Algorithms and Applications, 2009
A natural way to draw two planar graphs whose vertex sets are matched is to assign each matched pair a unique y-coordinate. In this paper we introduce the concept of such matched drawings, which are a relaxation of simultaneous geometric embeddings with mapping. We study which classes of graphs allow matched drawings and show that (i) two 3-connected planar graphs or a 3-connected planar graph and a tree may not be matched drawable, while (ii) two trees or a planar graph and a sufficiently restricted planar graph-such as an unlabeled level planar (ULP) graph or a graph of the family of "carousel graphs"-are always matched drawable. * Research partially supported by the MIUR Project "MAINSTREAM: Algorithms for massive information structures and data streams"