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198
Planar Orientations with Low OutDegree and Compaction of Adjacency Matrices
 Theoretical Computer Science
, 1991
"... We consider the problem of orienting the edges of a planar graph in such a way that the outdegree of each vertex is minimized. If, for each vertex v, the outdegree is at most d, then we say that such an orientation is dbounded. We prove the following results: ffl Each planar graph has a 5bounde ..."
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Cited by 34 (3 self)
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We consider the problem of orienting the edges of a planar graph in such a way that the outdegree of each vertex is minimized. If, for each vertex v, the outdegree is at most d, then we say that such an orientation is dbounded. We prove the following results: ffl Each planar graph has a 5bounded acyclic orientation, which can be constructed in linear time. ffl Each planar graph has a 3bounded orientation, which can be constructed in linear time. ffl A 6bounded acyclic orientation, and a 3bounded orientation, of each planar graph can each be constructed in parallel time O(log n log n) on an EREW PRAM, using O(n= log n log n) processors. As an application of these results, we present a data structure such that each entry in the adjacency matrix of a planar graph can be looked up in constant time. The data structure uses linear storage, and can be constructed in linear time. Department of Mathematics and Computer Science, University of California, Riverside, CA 92521. On...
Optimal upward planarity testing of singlesource digraphs
 SIAM Journal on Computing
, 1998
"... Abstract. A digraph is upward planar if it has a planar drawing such that all the edges are monotone with respect to the vertical direction. Testing upward planarity and constructing upward planar drawings is important for displaying hierarchical network structures, which frequently arise in softwar ..."
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Cited by 34 (4 self)
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Abstract. A digraph is upward planar if it has a planar drawing such that all the edges are monotone with respect to the vertical direction. Testing upward planarity and constructing upward planar drawings is important for displaying hierarchical network structures, which frequently arise in software engineering, project management, and visual languages. In this paper we investigate upward planarity testing of singlesource digraphs; we provide a new combinatorial characterization of upward planarity and give an optimal algorithm for upward planarity testing. Our algorithm tests whether a singlesource digraph with n vertices is upward planar in O(n) sequential time, and in O(log n) time on a CRCW PRAM with n log log n / log n processors, using O(n) space. The algorithm also constructs an upward planar drawing if the test is successful. The previously known best result is an O(n2)time algorithm by Hutton and Lubiw [Proc. 2nd ACM–SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 1991, pp. 203–211]. No efficient parallel algorithms for upward planarity testing were previously known.
Maximum Planar Subgraphs and Nice Embeddings: Practical Layout Tools
 ALGORITHMICA
, 1996
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Planar Minimally Rigid Graphs and PseudoTriangulations
, 2003
"... Pointed pseudotriangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under cer ..."
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Cited by 30 (14 self)
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Pointed pseudotriangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (incident to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide—to the best of our knowledge—the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces.
Pathbased depthfirst search for strong and biconnected components
 Information Processing Letters
, 2000
"... Key words: Graph, depthfirst search, strongly connected component, biconnected component, stack. ..."
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Cited by 30 (0 self)
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Key words: Graph, depthfirst search, strongly connected component, biconnected component, stack.
Computing crossing numbers in quadratic time
 J. Comput. Syst. Sci
, 2004
"... We show that for every fixed k ≥ 0 there is a quadratic time algorithm that decides whether a given graph has crossing number at most k and, if this is the case, computes a drawing of the graph in the plane with at most k crossings. 1. ..."
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Cited by 29 (0 self)
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We show that for every fixed k ≥ 0 there is a quadratic time algorithm that decides whether a given graph has crossing number at most k and, if this is the case, computes a drawing of the graph in the plane with at most k crossings. 1.
On the cutting edge: Simplified O(n) planarity by edge addition
 Journal of Graph Algorithms and Applications
, 2004
"... www.cs.uvic.ca/˜wendym ..."
Simultaneous embedding of planar graphs with few bends
 In 12th Symposium on Graph Drawing (GD
, 2004
"... 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, wit ..."
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Cited by 26 (6 self)
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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 crossingfree. 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 crossingsfree, 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 fixededges. 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 treepath pairs with at most one bend per treeedge 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).
Every minorclosed property of sparse graphs is testable
, 2007
"... Suppose G is a graph of bounded degree d, and one needs to remove ɛn of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G ′. In fact, a sim ..."
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Cited by 26 (3 self)
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Suppose G is a graph of bounded degree d, and one needs to remove ɛn of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G ′. In fact, a similar result is proved for any minorclosed property of bounded degree graphs. As an immediate corollary of the above result we infer that many well studied graph properties, like being planar, outerplanar, seriesparallel, bounded genus, bounded treewidth and several others, are testable with a constant number of queries. None of these properties was previously known to be testable even with o(n) queries. 1
NPCompleteness Results for Minimum Planar Spanners
"... For any fixed parameter t _> 1, a tspanner of a graph G is a spanning subgraph in which the distance between every pair of vertices is at most t times their distance in G. A minimum tspanner is a tspanner with minimum total edge weight or, in unweighted graphs, minimum number of edges. In this ..."
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Cited by 25 (0 self)
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For any fixed parameter t _> 1, a tspanner of a graph G is a spanning subgraph in which the distance between every pair of vertices is at most t times their distance in G. A minimum tspanner is a tspanner with minimum total edge weight or, in unweighted graphs, minimum number of edges. In this paper, we prove the AlPhardness of finding minimum tspanners for planar weighted graphs and digraphs if t _> 3, and for planar unweighted graphs and digraphs if t _> 5. We thus extend results on that problem to the interesting case where the instances are known to be planar. We also introduce the related problem of finding minimum planar tspanners and establish its Alphardness for similar fixed values of t.