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76
Testing Planarity of Partially Embedded Graphs
, 2009
"... We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in man ..."
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Cited by 27 (11 self)
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We study the following problem: Given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes hard an otherwise easy problem, we show that the planarity question remains polynomialtime solvable. Our algorithm is based on several combinatorial lemmata which show that the planarity of partially embedded graphs meets the “oncas” behaviour – obvious necessary conditions for planarity are also sufficient. These conditions are expressed in terms of the interplay between (a) rotation schemes and containment relationships between cycles and (b) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently. Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we improve our algorithm to reach lineartime complexity. Finally, we consider several generalizations of the problem, e.g. minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NPhard. Also, we show how our algorithm can be applied to solve related Graph Drawing problems.
Consistent labeling of rotating maps
 Int. Symp. Algorithms & Data Structures (WADS’11), volume 6844 of LNCS
, 2011
"... Abstract. Dynamic maps that allow continuous map rotations, e.g., on mobile devices, encounter new issues unseen in static map labeling before. We study the following dynamic map labeling problem: The input is a static, labeled map, i.e., a set P of points in the plane with attached nonoverlapping ..."
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Cited by 7 (3 self)
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Abstract. Dynamic maps that allow continuous map rotations, e.g., on mobile devices, encounter new issues unseen in static map labeling before. We study the following dynamic map labeling problem: The input is a static, labeled map, i.e., a set P of points in the plane with attached nonoverlapping horizontal rectangular labels. The goal is to find a consistent labeling of P under rotation that maximizes the number of visible labels for all rotation angles such that the labels remain horizontal while the map is rotated. A labeling is consistent if a single active interval of angles is selected for each label such that labels neither intersect each other nor occlude points in P at any rotation angle. We first introduce a general model for labeling rotating maps and derive basic geometric properties of consistent solutions. We show NPcompleteness of the active interval maximization problem even for unitsquare labels. We then present a constantfactor approximation for this problem based on line stabbing, and refine it further into an efficient polynomialtime approximation scheme (EPTAS). Finally, we extend the EPTAS to the more general setting of rectangular labels of bounded size and aspect ratio. 1
Computing large matchings fast
 TRANSACTIONS ON ALGORITHMS
"... In this paper we present algorithms for computing large matchings in 3regular graphs, graphs with maximum degree 3, and 3connected planar graphs. The algorithms give a guarantee on the size of the computed matching and take linear or slightly superlinear time. Thus they are faster than the bestkn ..."
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Cited by 2 (1 self)
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In this paper we present algorithms for computing large matchings in 3regular graphs, graphs with maximum degree 3, and 3connected planar graphs. The algorithms give a guarantee on the size of the computed matching and take linear or slightly superlinear time. Thus they are faster than the bestknown algorithm for computing maximum matchings in general graphs, which runs in O ( √ nm) time, where n denotes the number of vertices and m the number of edges of the given graph. For the classes of 3regular graphs and graphs with maximum degree 3 the bounds we achieve are known to be best possible. We also investigate graphs with block trees of bounded degree, where the dblock tree is the adjacency graph of the dconnected components of the given graph. In 3regular graphs and 3connected planar graphs with boundeddegree 2 and 4block trees, respectively, we show how to compute maximum matchings in slightly superlinear time.
TimeDependent SHARCRouting
 In Proceedings of the 16th Annual European Symposium on Algorithms (ESA’08
, 2008
"... In recent years, many speedup techniques for Dijkstra’s algorithm have been developed that make the computation of shortest paths in static road networks a matter of microseconds. However, only few of those techniques work in timedependent networks which, unfortunately, appear quite frequently in ..."
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Cited by 17 (9 self)
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In recent years, many speedup techniques for Dijkstra’s algorithm have been developed that make the computation of shortest paths in static road networks a matter of microseconds. However, only few of those techniques work in timedependent networks which, unfortunately, appear quite frequently in reality: Roads are predictably congested by traffic jams, and efficient timetable information systems rely on timedependent networks. Hence, a fast technique for routing in such networks is needed. In this work, we present an efficient timedependent route planning algorithm. It is based on our recently introduced SHARC algorithm, which we adapt by augmenting its basic ingredients such that correctness can still be guaranteed in a timedependent scenario. As a result, we are able to efficiently compute exact shortest paths in timedependent continentalsized transportation networks, both of roads and of railways. It should be noted that timedependent SHARC was the first efficient algorithm for timedependent route planning. 1
ManhattanGeodesic Embedding of Planar Graphs
"... In this paper, we explore a new convention for drawing graphs, the (Manhattan) geodesic drawing convention. It requires that edges are drawn as interiordisjoint monotone chains of axisparallel line segments, that is, as geodesics with respect to the Manhattan metric. First, we show that geodesic ..."
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Cited by 8 (2 self)
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In this paper, we explore a new convention for drawing graphs, the (Manhattan) geodesic drawing convention. It requires that edges are drawn as interiordisjoint monotone chains of axisparallel line segments, that is, as geodesics with respect to the Manhattan metric. First, we show that geodesic embeddability on the grid is equivalent to 1bend embeddability on the grid. For the latter question an efficient algorithm has been proposed. Second, we consider geodesic pointset embeddability where the task is to decide whether a given graph can be embedded on a given point set. We show that this problem is N Phard. In contrast, we efficiently solve geodesic polygonization—the special case where the graph is a cycle. Third, we consider geodesic pointset embeddability where the vertex–point correspondence is given. We show that on the grid, this problem is NPhard even for perfect matchings, but without the grid restriction, we solve the matching problem efficiently.
Forkforests in bicolored complete bipartite graphs
, 2012
"... Motivated by the problem in [6], which studies the relative efficiency of propositional proof systems, 2edge colorings of complete bipartite graphs are investigated. It is shown that if the edges of G = Kn,n are colored with black and white such that the number of black edges differs from the numbe ..."
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Motivated by the problem in [6], which studies the relative efficiency of propositional proof systems, 2edge colorings of complete bipartite graphs are investigated. It is shown that if the edges of G = Kn,n are colored with black and white such that the number of black edges differs from the number of white edges by at most 1, then there are at least n(1 − 1 / √ 2) vertexdisjoint forks with centers in the same partite set of G. Here, a fork is a graph formed by two adjacent edges of different colors. The bound is sharp. Moreover, an algorithm running in time O(n 2 log n √ nα(n 2, n) log n) and giving a largest such fork forest is found.
Forkforests in bicolored complete bipartite graphs
, 2013
"... Motivated by the problem in [6], which studies the relative efficiency of propositional proof systems, 2edge colorings of complete bipartite graphs are investigated. It is shown that if the edges of G = Kn,n are colored with black and white such that the number of black edges differs from the numbe ..."
Abstract
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Motivated by the problem in [6], which studies the relative efficiency of propositional proof systems, 2edge colorings of complete bipartite graphs are investigated. It is shown that if the edges of G = Kn,n are colored with black and white such that the number of black edges differs from the number of white edges by at most 1, then there are at least n(1 − 1/√2) vertexdisjoint forks with centers in the same partite set of G. Here, a fork is a graph formed by two adjacent edges of different colors. The bound is sharp. Moreover, an algorithm running in time O(n2 log n nα(n2, n) log n) and giving a largest such fork forest is found.
1 Executive Summary
"... This report documents the program and the outcomes of Dagstuhl Seminar 11191 “Graph Drawing with Algorithm Engineering Methods”. We summarize the talks, open problems, and working group discussions. ..."
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This report documents the program and the outcomes of Dagstuhl Seminar 11191 “Graph Drawing with Algorithm Engineering Methods”. We summarize the talks, open problems, and working group discussions.
Proportional Contact Representations of Planar Graphs
"... Abstract. We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by a pointcontact or a sidecontact between the corresponding polygons. Specifically, we consider proportional contact representations, where prespecified vertex w ..."
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Cited by 7 (6 self)
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Abstract. We study contact representations for planar graphs, with vertices represented by simple polygons and adjacencies represented by a pointcontact or a sidecontact between the corresponding polygons. Specifically, we consider proportional contact representations, where prespecified vertex weights must be represented by the areas of the corresponding polygons. Several natural optimization goals for such representations include minimizing the complexity of the polygons, the cartographic error, and the unused area. We describe constructive algorithms for proportional contact representations with optimal complexity for general planar graphs and planar 2segment graphs, which include maximal outerplanar graphs and partial 2trees. 1
Results 1  10
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76