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14
Orderly Spanning Trees with Applications to Graph Encoding and Graph Drawing
 In 12 th Symposium on Discrete Algorithms (SODA
, 2001
"... The canonical ordering for triconnected planar graphs is a powerful method for designing graph algorithms. This paper introduces the orderly pair of connected planar graphs, which extends the concept of canonical ordering to planar graphs not required to be triconnected. Let G be a connected planar ..."
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Cited by 33 (6 self)
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The canonical ordering for triconnected planar graphs is a powerful method for designing graph algorithms. This paper introduces the orderly pair of connected planar graphs, which extends the concept of canonical ordering to planar graphs not required to be triconnected. Let G be a connected planar graph. We give a lineartime algorithm that obtains an orderly pair (H
Simultaneous embedding of a planar graph and its dual on the grid
, 2002
"... Abstract. Traditional representations of graphs and their duals suggest the requirement that the dual vertices be placed inside their corresponding primal faces, and the edges of the dual graph cross only their corresponding primal edges. We consider the problem of simultaneously embedding a planar ..."
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Cited by 19 (10 self)
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Abstract. Traditional representations of graphs and their duals suggest the requirement that the dual vertices be placed inside their corresponding primal faces, and the edges of the dual graph cross only their corresponding primal edges. We consider the problem of simultaneously embedding a planar graph and its dual into a small integer grid such that the edges are drawn as straightline segments and the only crossings are between primaldual pairs of edges. We provide a lineartime algorithm that simultaneously embeds a 3connected planar graph and its dual on a (2n −2) ×(2n −2) integer grid, where n is the total number of vertices in the graph and its dual. Furthermore our embedding algorithm satisfies the two natural requirements mentioned above.
Orderly Spanning Trees with Applications
 SIAM Journal on Computing
, 2005
"... Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any c ..."
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Cited by 18 (3 self)
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Abstract. We introduce and study orderly spanning trees of plane graphs. This algorithmic tool generalizes canonical orderings, which exist only for triconnected plane graphs. Although not every plane graph admits an orderly spanning tree, we provide an algorithm to compute an orderly pair for any connected planar graph G, consisting of an embedded planar graph H isomorphic to G, and an orderly spanning tree of H. We also present several applications of orderly spanning trees: (1) a new constructive proof for Schnyder’s realizer theorem, (2) the first algorithm for computing an areaoptimal 2visibility drawing of a planar graph, and (3) the most compact known encoding of a planar graph with O(1)time query support. All algorithms in this paper run in linear time.
Improved Compact Routing Tables for Planar Networks via Orderly Spanning Trees
 In: 8 th Annual International Computing & Combinatorics Conference (COCOON). Volume 2387 of LNCS
, 2002
"... We address the problem of designing compact routing tables for an unlabeled connected nnode planar network G. For each node r of G, the designer is given a routing spanning tree Tr of G rooted at r, which speci es the routes for sending packets from r to the rest of G. ..."
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Cited by 12 (3 self)
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We address the problem of designing compact routing tables for an unlabeled connected nnode planar network G. For each node r of G, the designer is given a routing spanning tree Tr of G rooted at r, which speci es the routes for sending packets from r to the rest of G.
Parallel Computation and Graphical Visualization of the Minimum Crossing Number of a Graph
, 1998
"... Finding the minimum crossing number of a graph is an interesting and challenging problem in graph theory and applied mathematics. Real world applications of this problem, such as circuit layout and network design, are becoming more and more important. This thesis presents a parallel algorithm for f ..."
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Cited by 3 (0 self)
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Finding the minimum crossing number of a graph is an interesting and challenging problem in graph theory and applied mathematics. Real world applications of this problem, such as circuit layout and network design, are becoming more and more important. This thesis presents a parallel algorithm for finding the minimum crossing number of a graph, based on the first sequential algorithm presented in [20]. This parallel algorithm was tested on various architectures and a comparison of the corresponding results is given, including running time, efficiency, and speedup. Implementation of the algorithm gives us an ability to verify conjectures proposed for various families of graphs, as well as apply the algorithm to real world applications. Another important aspect of the problem is ability to draw the solution on the 2 \Gamma D plane. This thesis gives an overview of graph drawing algorithms, starting with the famous result by F'ary, and presents a new algorithm for drawing complete graphs...
Generalizing the Shift Method for Rectangular Shaped Vertices with Visibility Constraints
, 2009
"... In this paper we present a generalization of the shift method algorithm [4, 6] to obtain a straightline grid drawing of a triconnected graph, where vertex representations have a certain specified size. We propose vertex representations having a rectangular shape. Additionally, one may demand mainta ..."
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Cited by 1 (0 self)
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In this paper we present a generalization of the shift method algorithm [4, 6] to obtain a straightline grid drawing of a triconnected graph, where vertex representations have a certain specified size. We propose vertex representations having a rectangular shape. Additionally, one may demand maintainance of the criterion of strong visibility, that is, any possible line segment connecting two adjacent vertices cannot cross another vertex' representation. We prove that the proposed method produces a straightline grid drawing of a graph in linear time with an area bound, that is only extended by the size of the rectangles, compared to the bound of the original algorithm.
Drawing Planar Clustered Graphs in 2.5 Dimensions
"... Clustering techniques have proven to be useful in reducing the complexity of large networks. We present a method for drawing a clustered graph in 2.5D, given the clusters of the graph are connected in a planar way. The main contribution is to develop a weighted version of the well known 2D straight ..."
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Cited by 1 (1 self)
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Clustering techniques have proven to be useful in reducing the complexity of large networks. We present a method for drawing a clustered graph in 2.5D, given the clusters of the graph are connected in a planar way. The main contribution is to develop a weighted version of the well known 2D straight line drawing algorithm of de Fraysseix, Pach and Pollack [5] for planar graphs. This version allows for thick vertices and ensures a constraint we define as “visibility ” between connected vertices. The algorithm has been implemented and experimentally evaluated. 1
Leftist Canonical Ordering
, 2009
"... Canonical ordering is an important tool in planar graph drawing and other applications. Although a lineartime algorithm to determine canonical orderings has been known for a while, it is rather complicated to understand and implement, and the output is not uniquely determined. We present a new appr ..."
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Canonical ordering is an important tool in planar graph drawing and other applications. Although a lineartime algorithm to determine canonical orderings has been known for a while, it is rather complicated to understand and implement, and the output is not uniquely determined. We present a new approach that is simpler and more intuitive, and that computes a newly defined leftist canonical ordering of a triconnected graph which is a uniquely determined leftmost canonical ordering.