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The Planar Slope Number of Planar Partial 3Trees of Bounded Degree
"... Abstract. It is known that every planar graph has a planar embedding where edges are represented by noncrossing straightline segments. We study the planar slope number, i.e., the minimum number of distinct edgeslopes in such a drawing of a planar graph with maximum degree ∆. We show that the pla ..."
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that the planar slope number of every planar partial 3tree and also every plane partial 3tree is at most O(∆5). In particular, we answer the question of Dujmovic ́ et al. [Computational Geometry 38 (3), pp. 194–212 (2007)] whether there is a function f such that plane maximal outerplanar graphs can be drawn
The Planar Slope Number of Planar Partial 3Trees of Bounded Degree
, 2010
"... It is known that every planar graph has a planar embedding where edges are represented by noncrossing straightline segments. We study the planar slope number, i.e., the minimum number of distinct edgeslopes in such a drawing of a planar graph with maximum degree ∆. We show that the planar slope n ..."
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number of every planar partial 3tree and also every plane partial 3tree is at most O(∆5). In particular, we answer the question of Dujmovic ́ et al. [Computational Geometry 38 (3), pp. 194–212 (2007)] whether there is a function f such that plane maximal outerplanar graphs can be drawn using at most f
The Planar Slope Number of Planar Partial 3Trees of Bounded Degree
, 2009
"... It is known that every planar graph has a planar embedding where edges are represented by noncrossing straightline segments. We study the planar slope number, i.e., the minimum number of distinct edgeslopes in such a drawing of a planar graph with maximum degree Δ. We show that the planar slope ..."
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Cited by 6 (0 self)
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number of every seriesparallel graph of maximum degree three is three. We also show that the planar slope number of every planar partial 3tree and also every plane partial 3tree is at most 2O(Δ). In particular, we answer the question of Dujmovic ́ et al. [Computational Geometry 38 (3), pp. 194
Drawing outer 1planar graphs with few slopes
, 2015
"... A graph is outer 1planar if it admits a drawing where each vertex is on the outer face and each edge is crossed by at most another edge. Outer 1planar graphs are a superclass of the outerplanar graphs and a subclass of the planar partial 3trees. We show that an outer 1planar graph G of bounded d ..."
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; the best known upper bound on the planar slope number of planar partial 3trees of bounded degree ∆ is O(∆5) as proved by Jeĺınek et al. [16].
A Separator Theorem for Planar Graphs
, 1977
"... Let G be any nvertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than 2& & vertices. We exhibit an algorithm which ..."
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Cited by 465 (1 self)
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Let G be any nvertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2n/3 vertices, and C contains no more than 2& & vertices. We exhibit an algorithm which
Probabilistic Roadmaps for Path Planning in HighDimensional Configuration Spaces
 IEEE INTERNATIONAL CONFERENCE ON ROBOTICS AND AUTOMATION
, 1996
"... A new motion planning method for robots in static workspaces is presented. This method proceeds in two phases: a learning phase and a query phase. In the learning phase, a probabilistic roadmap is constructed and stored as a graph whose nodes correspond to collisionfree configurations and whose edg ..."
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Cited by 1276 (124 self)
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to be relatively easy to choose, Increased efficiency can also be achieved by tailoring some components of the method (e.g., the local planner) to the considered robots. In this paper the method is applied to planar articulated robots with many degrees of freedom. Experimental results show that path planning can
Efficient planarity testing
 J. ASSOC. COMPUT. MACH
, 1974
"... This paper describes an efficient algorithm to determine whether an arbitrary graph G can be embedded in the plane. The algorithm may be viewed as an iterative version of a method originally proposed by Auslander and Parter and correctly formulated by Goldstein. The algorithm uses depthfirst sear ..."
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Cited by 280 (5 self)
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first search and has O(V) time and space bounds, where V is the number of vertices in G. An ALGOS implementation of the algorithm successfully tested graphs with as many as 900 vertices in less than 12 seconds.
Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations
, 2005
"... How do real graphs evolve over time? What are “normal” growth patterns in social, technological, and information networks? Many studies have discovered patterns in static graphs, identifying properties in a single snapshot of a large network, or in a very small number of snapshots; these include hea ..."
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Cited by 534 (48 self)
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How do real graphs evolve over time? What are “normal” growth patterns in social, technological, and information networks? Many studies have discovered patterns in static graphs, identifying properties in a single snapshot of a large network, or in a very small number of snapshots; these include
A fast and high quality multilevel scheme for partitioning irregular graphs
 SIAM JOURNAL ON SCIENTIFIC COMPUTING
, 1998
"... Recently, a number of researchers have investigated a class of graph partitioning algorithms that reduce the size of the graph by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph [Bui and Jones, Proc. ..."
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Cited by 1173 (16 self)
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Recently, a number of researchers have investigated a class of graph partitioning algorithms that reduce the size of the graph by collapsing vertices and edges, partition the smaller graph, and then uncoarsen it to construct a partition for the original graph [Bui and Jones, Proc.
Results 1  10
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132,560