Algorithms for laying out large graphs have seen significant progress in the past decade. However, browsing large graphs remains a challenge. Rendering thousands of graphical elements at once often results in a cluttered image, and navigating these elements naively can cause disorientation. To address this challenge we propose a method called GraphMaps, mimicking the browsing experience of online geographic maps. GraphMaps creates a sequence of layers, where each layer refines the previous one. During graph browsing, GraphMaps chooses the layer corresponding to the zoom level, and renders only those entities of the layer that intersect the current viewport. The result is that, regardless of the graph size, the number of entities rendered at each view does not exceed a predefined threshold, yet all graph elements can be explored by the standard zoom and pan operations. GraphMaps preprocesses a graph in such a way that during browsing, the geometry of the entities is stable, and the viewer is responsive. Our case studies indicate that GraphMaps is useful in gaining an overview of a large graph, and also in exploring a graph on a finer level of detail.
Abstract. We present a simple and versatile formulation of grid-based graph representation problems as an integer linear program (ILP) and a corresponding SAT instance. In a grid-based representation vertices and edges correspond to axisparallel boxes on an underlying integer grid; boxes can be further constrained in their shapes and interactions by additional problem-specific constraints. We describe a general d-dimensional model for grid representation problems. This model can be used to solve a variety of NP-hard graph problems, including pathwidth, bandwidth, optimum st-orientation, area-minimal (bar-k) visibility representation, boxicity-k graphs and others. We implemented SAT-models for all of the above problems and evaluated them on the Rome graphs collection. The experiments show that our model successfully solves NP-hard problems within few minutes on small to medium-size Rome graphs.
Abstract. Greedy embedding (or drawing) is a simple and efficient strategy to route messages in wireless sensor networks. For each source-destination pair of nodes s, t in a greedy embedding there is always a neighbor u of s that is closer to t according to some distance metric. The existence of greedy embeddings in the Euclidean plane R 2 is known for certain graph classes such as 3-connected planar graphs. We completely characterize the trees that admit a greedy embedding in R 2 . This answers a question by Angelini et al. (Graph Drawing 2009) and is a further step in characterizing the graphs that admit Euclidean greedy embeddings.
An st-path in a drawing of a graph is self-approaching if during the traversal of the corresponding curve from s to any point t on the curve the distance to t is non-increasing. A path has increasing chords if it is self-approaching in both directions. A drawing is self-approaching (increasing-chord) if any pair of vertices is connected by a self-approaching (increasing-chord) path.We study self-approaching and increasing-chord drawings of triangulations and 3-connected planar graphs. We show that in the Euclidean plane, triangulations admit increasing-chord drawings, and for planar 3-trees we can ensure planarity. We prove that strongly monotone (and thus increasing-chord) drawings of trees and binary cactuses require exponential resolution in the worst case, answering an open question by Kindermann et al. [13]. Moreover, we provide a binary cactus that does not admit a self-approaching drawing. Finally, we show that 3-connected planar graphs admit increasing-chord drawings in the hyperbolic plane and characterize the trees that admit such drawings. * A preliminary version of this paper appeared at the 22nd International Symposium on Graph Drawing, geodesic-path tendency have been suggested, namely greedy drawings [20], (strongly) monotone drawings [2], and self-approaching and increasing-chord drawings [1]. Note that throughout this paper, all drawings are straight-line and vertices are mapped to distinct points.The notion of greedy drawings came first and was introduced by Rao et al. [20]. Motivated by greedy routing schemes, e.g., for sensor networks, one seeks a drawing where for every pair of vertices s and t, there exists an st-path, along which the distance to t decreases in every vertex. This ensures that greedily sending a message to a vertex that is closer to the destination guarantees delivery. Papadimitriou and Ratajczak conjectured that every 3-connected planar graph admits a greedy embedding into the Euclidean plane [18]. This conjecture has been proved independently by Leighton and Moitra [16] and Angelini et al. [5]. Kleinberg [14] showed that every connected graph has a greedy drawing in the hyperbolic plane. Eppstein and Goodrich [8] showed how to construct such an embedding, in which the coordinates of each vertex are represented using only O(log n) bits, and Goodrich and Strash [10] provided a corresponding succinct representation for greedy embeddings of 3-connected planar graphs in R 2 . Angelini et al. [3] showed that some graphs require exponential area for a greedy drawing in R 2 . Wang and He [23] used a custom distance metric to construct planar, convex and succinct greedy embeddings of 3-connected planar graphs using Schnyder realizers [22]. Nöllenburg and Prutkin [17] characterized trees admitting a Euclidean greedy embedding. However, a number of interesting questions remain open, e.g., whether every 3-connected planar graph admits a planar and convex Euclidean greedy embedding (strong Papadimitriou-Ratajczak conjecture [18]). Regarding planar greedy drawings of triangulations, the...
In wireless ad hoc or sensor networks, distributed node coloring is a fundamental problem closely related to establishing efficient communication through TDMA schedules. For networks with maximum degree ∆, a ∆ + 1 coloring is the ultimate goal in the distributed setting as this is always possible. In this work we propose ∆ + 1 coloring algorithms for the synchronous and asynchronous setting. All algorithms have a runtime of O(∆ log n) time slots. This improves on the previous algorithms for the SINR model either in terms of the number of required colors or the runtime and matches the runtime of local broadcasting in the SINR model (which can be seen as a lower bound).
Disk contact representations realize graphs by mapping vertices bijectively to interior-disjoint disks in the plane such that two disks touch each other if and only if the corresponding vertices are adjacent in the graph. Deciding whether a vertex-weighted planar graph can be realized such that the disks' radii coincide with the vertex weights is known to be NP-hard. In this work, we reduce the gap between hardness and tractability by analyzing the problem for special graph classes. We show that it remains NP-hard for outerplanar graphs with unit weights and for stars with arbitrary weights, strengthening the previous hardness results. On the positive side, we present constructive linear-time recognition algorithms for caterpillars with unit weights and for embedded stars with arbitrary weights.
We study contact representations of edge-weighted planar graphs, where vertices are represented as interior-disjoint rectangles or rectilinear polygons and edges are represented as contacts of vertex boundaries whose contact lengths represent the edge weights.For the case of rectangles, we show that, for any given edge-weighted planar graph whose outer face is a quadrangle, that is internally triangulated and that has no separating triangles, we can construct in linear time an edge-proportional rectangular dual (contact lengths are equal to the given edge weights and the union of all rectangles is again a rectangle) or report failure if none exists. In the case of arbitrarily many outer vertices, we show that deciding whether a square layout exists is NP-complete. If the orientation of each contact is specified by a so-called regular edge labeling and edge weights are lower bounds on the contact lengths, a corresponding rectangular dual that minimizes the area and perimeter of the enclosing rectangle can be found in linear time. On the other hand, without a given regular edge labeling, the same problem is NP-complete, as is the question whether a rectangular dual exists given lower and upper bounds on the contact lengths.For the case of rectilinear polygons, we give a complete characterization of the polygon complexity required for representing connected internally triangulated graphs: For outerplanar graphs and graphs with a single inner vertex polygon, complexity 8 is sufficient and necessary, and for graphs with two adjacent or multiple non-adjacent internal vertices the required polygon complexity is unbounded.
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