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Implementing a Partitioned 2-Page Book Embedding Testing Algorithm
No full textIn a book embedding the vertices of a graph are placed on the "spine" of a "book" and the edges are assigned to "pages" so that edges on the same page do not cross. In the Partitioned 2-page Book Embedding problem egdes are partitioned into two sets and , the pages are two, the edges of are assigned to page 1, and the edges of are assigned to page 2. The problem consists of checking if an ordering of the vertices exists along the spine so that the edges of each page do not cross. Hong and Nagamochi [13] give an interesting and complex linear time algorithm for tackling Partitioned 2-page Book Embedding based on SPQR-trees. We show an efficient implementation of this algorithm and show its effectiveness by performing a number of experimental tests. Because of the relationships [13] between Partitioned 2-page Book Embedding and clustered planarity we yield as a side effect an implementation of a clustered planarity testing where the graph has exactly two clusters
Upward Planarity Testing via SAT
No full textA directed acyclic graph is upward planar if it allows a drawing without edge crossings where all edges are drawn as curves with monotonously increasing y-coordinates. The problem to decide whether a graph is upward planar or not is NP-complete in general, and while special graph classes are polynomial time solvable, there is not much known about solving the problem for general graphs in practice. The only attempt so far was a branch-and-bound algorithm over the graph’s triconnectivity structure which was able to solve sparse graphs.
In this paper, we propose a fundamentally different approach, based on the seemingly novel concept of ordered embeddings. We carefully model the problem as a special SAT instance, i.e., a logic formula for which we check satisfiability. Solving these SAT instances allows us to decide upward planarity for arbitrary graphs. We then show experimentally that this approach seems to dominate the known alternative approaches and is able to solve traditionally used graph drawing benchmarks effectively
Testing Maximal 1-Planarity of Graphs with a Rotation System in Linear Time
No full textA 1-planar graph is a graph that can be embedded in the plane with at most one crossing per edge. It is known that testing 1-planarity of a graph is NP-complete. A 1-planar embedding of a graph G is maximal if no edge can be added without violating the 1-planarity of G. In this paper we show that the problem of testing maximal 1-planarity of a graph G can be solved in linear time, if a rotation system (i.e., the circular ordering of edges for each vertex) is given. We also prove that there is at most one maximal 1-planar embedding of G that preserves the given rotation system, and our algorithm produces such an embedding in linear time, if it exists
Visualizing Streaming Text Data with Dynamic Graphs and Maps
No full textThe many endless rivers of text now available present a serious challenge in the task of gleaning, analyzing and discovering useful information. In this paper, we describe a methodology for visualizing text streams in real-time modeled as a dynamic graph and its derived map. The approach automatically groups similar messages into “countries,” with keyword summaries, using semantic analysis, graph clustering and map generation techniques. It handles the need for visual stability across time by dynamic graph layout and Procrustes projection techniques, enhanced with a novel stable component packing algorithm. The result provides a continuous, succinct view of evolving topics of interest. To make these ideas concrete, we describe their application to an online service called TwitterScope
On the Usability of Lombardi Graph Drawings
No full textA recent line of work in graph drawing studies Lombardi drawings, i.e., drawings with circular-arc edges and perfect angular resolution at vertices. Little is known about the effects of curved edges versus straight edges in typical graph reading tasks. In this paper we present the first user evaluation that empirically measures the readability of three different layout algorithms (traditional spring embedder and two recent near-Lombardi force-based algorithms) for three different tasks (shortest path, common neighbor, vertex degree). The results indicate that, while users prefer the Lombardi drawings, the performance data do not present such a positive picture
Planar Lombardi Drawings of Outerpaths
No full textIntroduction
A Lombardi drawing of a graph is a drawingwhere edges are represented by circular arcs that meet at each vertex v with perfect angular resolution 360°/deg(v) [3]. It is known that Lombardi drawings do not always exist, and in particular, that planar Lombardi drawings of planar graphs do not always exist [1], even when the embedding is not fixed. Existence of planar Lombardi drawings is known for restricted classes of graphs, such as subcubic planar graphs [4], trees [2], Halin graphs and some very symmetric planar graphs [3]. On the other hand, all 2-degenerate graphs, including all outerplanar graphs, have Lombardi drawings, but not necessarily planar ones [3]. One question that was left open is whether outerplanar graphs always have planar Lombardi drawings or not.
In this note, we report that the answer is “yes” for a more restricted subclass: the outerpaths, i.e., outerplanar graphs whose weak dual is a path. We sketch an algorithm that produces an outerplanar Lombardi drawing of any outerpath, in linear time
Strongly-Connected Outerplanar Graphs with Proper Touching Triangle Representations
No full textA proper touching triangle representation R of an n-vertex planar graph consists of a triangle divided into n non-overlapping triangles. A pair of triangles are considered to be adjacent if they share a partial side of positive length. Each triangle in R represents a vertex, while each pair of adjacent triangles represents an edge in the planar graph. We consider the problem of determining when a proper touching triangle representation exists for a strongly-connected outerplanar graph, which is biconnected and after the removal of all degree-2 vertices and outeredges, the resulting connected subgraph only has chord edges (w.r.t. the original graph). We show that such a graph has a proper representation if and only if the graph has at most two internal faces (i.e., faces with no outeredges)
Drawing Permutations with Few Corners
No full textA permutation may be represented by a collection of paths in the plane. We consider a natural class of such representations, which we call tangles, in which the paths consist of straight segments at 45 degree angles, and the permutation is decomposed into nearest-neighbour transpositions. We address the problem of minimizing the number of crossings together with the number of corners of the paths, focusing on classes of permutations in which both can be minimized simultaneously. We give algorithms for computing such tangles for several classes of permutations
Progress on Partial Edge Drawings
No full textRecently, a new way of avoiding crossings in straight-line drawings of non-planar graphs has been investigated. The idea of partial edge drawings (PED) is to drop the middle part of edges and rely on the remaining edge parts called stubs. We focus on a symmetric model (SPED) that requires the two stubs of an edge to be of equal length. In this way, the stub at the other endpoint of an edge assures the viewer of the edge's existence. We also consider an additional homogeneity constraint that forces the stub lengths to be a given fraction δ of the edge lengths (δ-SHPED). Given length and direction of a stub, this model helps to infer the position of the opposite stub.
We show that, for a fixed stub---edge length ratio δ, not all graphs have a δ-SHPED. Specifically, we show that does not have a 1/4-SHPED, while bandwidth-k graphs always have a -SHPED. We also give bounds for complete bipartite graphs. Further, we consider the problem MaxSPED where the task is to compute the SPED of maximum total stub length that a given straight-line drawing contains. We present an efficient solution for 2-planar drawings and a 2-approximation algorithm for the dual problem
Planar Graphs as VPG-Graphs
No full textA graph is when it has an intersection representation by paths in a rectangular grid with at most k bends (turns). It is known that all planar graphs are and this was conjectured to be tight. We disprove this conjecture by showing that all planar graphs are . We also show that the 4-connected planar graphs are a subclass of the intersection graphs of Z-shapes (i.e., a special case of ). Additionally, we demonstrate that a representation of a planar graph can be constructed in time. We further show that the triangle-free planar graphs are contact graphs of: L-shapes, Γ-shapes, vertical segments, and horizontal segments (i.e., a special case of contact ). From this proof we gain a new proof that bipartite planar graphs are a subclass of 2-DIR