We consider the problem of simultaneous embedding of planar graphs. There are two variants

We consider several variations of the simultaneous embedding problem for planar graphs. We begin with a simple proof that not all pairs of planar graphs have simultaneous geometric embedding. However, using bends, pairs of planar graphs can be simultaneously embedded on the O(n 2) × O(n 2) grid, with at most three bends per edge, where n is the number of vertices. The O(n) time algorithm guarantees that two corresponding vertices in the graphs are mapped to the same location in the final drawing and that both the drawings are crossing-free. The special case when both input graphs are trees has several applications, such as contour tree simplification and evolutionary biology. We show that if both the input graphs are are trees, only one bend per edge is required. The O(n) time algorithm guarantees that both drawings are crossings-free, corresponding tree vertices are mapped to the same locations, and all vertices (and bends) are on the O(n 2) × O(n 2) grid (O(n 3) × O(n 3) grid). For the special case when one of the graphs is a tree and the other is a path we can find simultaneous embedding with fixed-edges. That is, we can guarantee that corresponding vertices are mapped to the same locations and that corresponding edges are drawn the same way. We describe an O(n) time algorithm for simultaneous embedding with fixededges for tree-path pairs with at most one bend per tree-edge and no bends along path edges, such that all vertices (and bends) are on the O(n) × O(n 2) grid, (O(n 2) × O(n 3) grid).

Abstract. In this paper we consider the problem of drawing and displaying a series of related graphs, i.e., graphs that share all, or parts of the same vertex set. We designed and implemented three different algorithms for simultaneous graphs drawing and three different visualization schemes. The algorithms are based on a modification of the force-directed algorithm that allows us to take into account vertex weights and edge weights in order to achieve mental map preservation while obtaining individually readable drawings. The implementation is in Java and the system can be downloaded at

What are the hottest computer science research topics today? Which research areas are experiencing steady decline? How many co-authors are typical for a research paper today and 20 years ago? Who are the most prolific writers? In this paper, we attempt to address these questions as well as study collaboration patterns, research communities, interactions between related research specialties, and the evolution of these characteristics through time. For our analysis we use data from the Association of Computing Machinery's Digital Library of Scientific Literature (ACM Portal) which contains over a hundred thousand research papers and authors. We use a novel technique for visualization of large graphs that evolve through time. Given a dynamic graph, the layout algorithm produces two-dimensional representations of each time-slice, while preserving the mental map of the graph from one slice to the next. A combined view, with all the time-slices can also be viewed and explored. Graphs with tens of thousands of vertices and edges, resulting from specific queries to our local copy of the ACM database, are generated and displayed in seconds. The images in this paper are produced by a graph layout tool which uses the dynamic graph layout algorithm.

Foundation (JU204/10-1). Abstract. We study the problem of simultaneously embedding several graphs on the same vertex set in such a way that edges common to two or more graphs are represented by the same curve. This problem is known as simultaneously embedding graphs with fixed edges. We show that this problem is closely related to the weak realizability problem: Can a graph be drawn such that all edge crossings occur in a given set of edge pairs? By exploiting this relationship we can explain why the simultaneous embedding problem is challenging, both from a computational and a combinatorial point of view. More precisely, we prove that simultaneously embedding graphs with fixed edges is NP-complete even for three planar graphs. For two planar graphs the complexity status is still open. 1

A geometric simultaneous embedding of two graphs G1 = (V1, E1) and G2 = (V2, E2) with a bijective mapping of their vertex sets γ: V1 → V2 is a pair of planar straight-line drawings Γ1 of G1 and Γ2 of G2, such that each vertex v2 = γ(v1) is mapped in Γ2 to the same point where v1 is mapped in Γ1, where v1 ∈ V1 and v2 ∈ V2. In this paper we examine several constrained versions and a relaxed version of the geometric simultaneous embedding problem. We show that if the input graphs are assumed to share no common edges this does not seem to yield large classes of graphs that can be simultaneously embedded. Further, if a prescribed combinatorial embedding for each input graph must be preserved, then we can answer some of the problems that are still open for geometric simultaneous embedding. Finally, we present some positive and negative results on the near-simultaneous embedding problem, in which vertices are not forced to be placed exactly in the same, but just in “near” points in different drawings.

Motivated by applications of social network analysis and of Web search clustering engines, we describe an algorithm and a system for the display and the visual analysis of two graphs G1 and G2 such that each Gi is defined on a different data set with its own primary relationships and there are secondary relationships between the vertices of G1 and those of G2. Our main goal is to compute a drawing of G1 and G2 that makes clearly visible the relations between the two graphs by avoiding their crossings, and that also takes into account some other important aesthetic requirements like number of bends, area, and aspect ratio. Application examples and experiments on the system performances are also presented.

Web site traffic analysis studies how different pages forming Web sites are accessed by their target audience over time. From a business intelligence point of view, it represents one of the key activities for many private companies and public institutions. Recent papers pointed out that Web site traffic analysis is especially useful if it focuses on the user interest into the relevant concepts described in a Web site rather than counting user accesses to the distinct pages forming the Web site. This paper extends existing measures of conceptual Web site traffic analysis and describes a new system that supports this analysis by means of graph visualization techniques. The graph drawing engine of the system is a force-directed heuristic that computes a simultaneous embedding of two suitably defined graphs, that are non-planar in general. The heuristic is especially designed to take into account two important aesthetic criteria of the drawing: crossing resolution and geodesic edge tendency. We also present some experiments and case studies to show the effectiveness of the proposed approach in practice. Submitted:

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