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280
Forcing Disjoint Segments in the Plane
"... Consider a geometric graph given by n points in the plane (in general position) and m line segments, each segment joining a pair of the given points. We show that: if m ≥ 3n + 1 then there are 3 pairwise disjoint segments; if m ≥ 10n + 1 then there are 4 disjoint segments; and if m ≥ ckn(log n) k−4 ..."
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Cited by 11 (0 self)
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Consider a geometric graph given by n points in the plane (in general position) and m line segments, each segment joining a pair of the given points. We show that: if m ≥ 3n + 1 then there are 3 pairwise disjoint segments; if m ≥ 10n + 1 then there are 4 disjoint segments; and if m ≥ ckn(log n) k
Binary Plane Partitions for Disjoint Line Segments
"... A binary space partition (BSP) for a set of disjoint objects in Euclidean space is a recursive decomposition, where each step partitions the space (and some of the objects) along a hyperplane and recurses on the objects clipped in each of the two open halfspaces. The size of a BSP is defined as the ..."
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Cited by 2 (1 self)
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as the number of resulting fragments of the input objects. It is shown that every set of n disjoint line segments in the plane admits a BSP of size O(n log n / log log n). This bound is best possible apart from the constant factor. 1
Disjoint Edges in Geometric Graphs
 DISCRETE COMPUT GEOM 4:287290 (1989)
, 1989
"... Answering an old question in combinatorial geometry, we show that any configuration consisting of a set V of n points in general position in the plane and a set of 6n 5 closed straight line segments whose endpoints lie in V, contains three pairwise disjoint line segments. ..."
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Cited by 16 (0 self)
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Answering an old question in combinatorial geometry, we show that any configuration consisting of a set V of n points in general position in the plane and a set of 6n 5 closed straight line segments whose endpoints lie in V, contains three pairwise disjoint line segments.
A Long Noncrossing Path Among Disjoint Segments in the Plane
, 2005
"... Let L be a collection of n pairwise disjoint segments in general position in the plane. We show that one can find a subcollection of Ω(n 1/3) segments that can be completed to a noncrossing simple path by adding rectilinear edges between endpoints of pairs of segments. On the other hand, there is ..."
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Let L be a collection of n pairwise disjoint segments in general position in the plane. We show that one can find a subcollection of Ω(n 1/3) segments that can be completed to a noncrossing simple path by adding rectilinear edges between endpoints of pairs of segments. On the other hand
Tangencies between families of disjoint regions in the plane
, 2010
"... Let C be a family of n convex bodies in the plane, which can be decomposed into k subfamilies of pairwise disjoint sets. It is shown that the number of tangencies between the members of C is at most O(kn), and that this bound cannot be improved. If we only assume that our sets are connected and vert ..."
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Cited by 1 (0 self)
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Let C be a family of n convex bodies in the plane, which can be decomposed into k subfamilies of pairwise disjoint sets. It is shown that the number of tangencies between the members of C is at most O(kn), and that this bound cannot be improved. If we only assume that our sets are connected
Alternating paths through disjoint line segments
 Inform. Process. Lett
, 2003
"... Abstract. We show that every segment endpoint visibility graph on Ò disjoint line segments in the plane admits an alternating path of length ¢ ÐÓ � Ò, answering a question of Bose. This bound is optimal apart from a constant factor. We also give bounds on the constants hidden by the asymptotic notat ..."
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Cited by 7 (1 self)
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Abstract. We show that every segment endpoint visibility graph on Ò disjoint line segments in the plane admits an alternating path of length ¢ ÐÓ � Ò, answering a question of Bose. This bound is optimal apart from a constant factor. We also give bounds on the constants hidden by the asymptotic
Disjoint Compatible Geometric Matchings
"... We show that the free space around n disjoint line segments in general position in the plane admits a decomposition into at most n + 1 convex cells such that the dual graph of the decomposition contains two edgedisjoint spanning trees. Furthermore, if n is even, then the segments are edges of a col ..."
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Cited by 5 (1 self)
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We show that the free space around n disjoint line segments in general position in the plane admits a decomposition into at most n + 1 convex cells such that the dual graph of the decomposition contains two edgedisjoint spanning trees. Furthermore, if n is even, then the segments are edges of a
Every set of disjoint line segments admits a binary tree
"... Given a set of n disjoint line segments in the plane, we show that it is always possible to form a tree with the endpoints of the segments such that each line segment is an edge of the tree, the tree has no crossing edges, and the maximum vertex degree of the tree is 3. Furthermore, there exist con ..."
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Cited by 19 (0 self)
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Given a set of n disjoint line segments in the plane, we show that it is always possible to form a tree with the endpoints of the segments such that each line segment is an edge of the tree, the tree has no crossing edges, and the maximum vertex degree of the tree is 3. Furthermore, there exist
On disjoint Borel uniformizations
 Advances in Mathematics
"... Abstract. Larman showed that any closed subset of the plane with uncountable vertical crosssections has ℵ1 disjoint Borel uniformizing sets. Here we show that Larman’s result is best possible: there exist closed sets with uncountable crosssections which do not have more than ℵ1 disjoint Borel unif ..."
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Cited by 4 (1 self)
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Abstract. Larman showed that any closed subset of the plane with uncountable vertical crosssections has ℵ1 disjoint Borel uniformizing sets. Here we show that Larman’s result is best possible: there exist closed sets with uncountable crosssections which do not have more than ℵ1 disjoint Borel
Kinetic Binary Space Partitions for Intersecting Segments and Disjoint Triangles (Extended Abstract)
, 1998
"... We describe randomized algorithms for efficiently maintaining a binary space partition of continuously moving, possibly intersecting, line segments in the plane, and of continuously moving but disjoint triangles in space. Our twodimensional BSP has depth O(log n) and size O(n log n + k) and can be ..."
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Cited by 20 (9 self)
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We describe randomized algorithms for efficiently maintaining a binary space partition of continuously moving, possibly intersecting, line segments in the plane, and of continuously moving but disjoint triangles in space. Our twodimensional BSP has depth O(log n) and size O(n log n + k) and can
Results 1  10
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280