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GRAPH DRAWINGS WITH FEW SLOPES
, 2006
"... The slopenumber of a graph G is the minimum number of distinct edge slopes in a straightline drawing of G in the plane. We prove that for ∆ ≥ 5 and all large n, there is a ∆regular nvertex graph with slopenumber at least 8+ε 1− n ∆+4. This is the best known lower bound on the slopenumber of a ..."
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Cited by 12 (3 self)
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The slopenumber of a graph G is the minimum number of distinct edge slopes in a straightline drawing of G in the plane. We prove that for ∆ ≥ 5 and all large n, there is a ∆regular nvertex graph with slopenumber at least 8+ε 1− n ∆+4. This is the best known lower bound on the slopenumber of a graph with bounded degree. We prove upper and lower bounds on the slopenumber of complete bipartite graphs. We prove a general upper bound on the slopenumber of an arbitrary graph in terms of its bandwidth. It follows that the slopenumber of interval graphs, cocomparability graphs, and ATfree graphs is at most a function of the maximum degree. We prove that graphs of bounded degree and bounded treewidth have slopenumber at most O(log n). Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered.
Combinatorial Geometry with Algorithmic Applications – The Alcala Lectures
"... These lecture notes are a compilation of surveys of the topics that are presented ..."
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Cited by 6 (0 self)
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These lecture notes are a compilation of surveys of the topics that are presented
Solution of Scott’s Problem on the Number of Directions Determined by a Point Set in 3Space, Discrete Comput
 Geom
"... Let P be a set of n points in R 3, not all in a common plane. We solve a problem of Scott (1970) by showing that the connecting lines of P assume at least 2n 5 dierent directions if n is odd and at least 2n 7 if n is even. The bound for odd n is sharp. 1 ..."
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Cited by 5 (1 self)
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Let P be a set of n points in R 3, not all in a common plane. We solve a problem of Scott (1970) by showing that the connecting lines of P assume at least 2n 5 dierent directions if n is odd and at least 2n 7 if n is even. The bound for odd n is sharp. 1
Blocking visibility for points in general position
"... For a finite set P in the plane, let b(P) be the smallest possible size of a set Q, Q ∩P = ∅, such that every segment with both endpoints in P contains at least one point of Q. We raise the problem of estimating b(n), the minimum of b(P) over all npoint sets P with no three points collinear. By a c ..."
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Cited by 2 (0 self)
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For a finite set P in the plane, let b(P) be the smallest possible size of a set Q, Q ∩P = ∅, such that every segment with both endpoints in P contains at least one point of Q. We raise the problem of estimating b(n), the minimum of b(P) over all npoint sets P with no three points collinear. By a construction similar to Behrend’s construction of sparse sets of integers avoiding 3term arithmetic progressions, we obtain the upper bound b(n) = ne O( √ log n). We have no superlinear lower bound, but for P in convex position, we show that b(P) = Ω(n log n).
Really straight drawings II: Nonplanar graphs
, 2005
"... We study straightline drawings of nonplanar graphs with few slopes. Interval graphs, cocomparability graphs and ATfree graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings ..."
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We study straightline drawings of nonplanar graphs with few slopes. Interval graphs, cocomparability graphs and ATfree graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings with O(log n) slopes. Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered.
On
"... the number of distinct directions of planes determined by n points in R3 ..."
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On the number of tetrahedra with minimum, unit, and distinct volumes in threespace ∗
, 710
"... We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by n points in 3space, and in general in d dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by n points in R3 is at most 2 3n3 − O(n2), and there are point sets for ..."
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We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by n points in 3space, and in general in d dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by n points in R3 is at most 2 3n3 − O(n2), and there are point sets for which this number is 3 16n3 − O(n2). We also present an O(n3) time algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby extend an algorithm of Edelsbrunner, O’Rourke, and Seidel. In general, for every k, d ∈ N, 1 ≤ k ≤ d, the maximum number of kdimensional simplices of minimum (nonzero) volume spanned by n points in R d is Θ(n k). (ii) The number of unitvolume tetrahedra determined by n points in R 3 is O(n 7/2), and there are point sets for which this number is Ω(n 3 log log n). (iii) For every d ∈ N, the minimum number of distinct volumes of all fulldimensional simplices determined by n points in R d, not all on a hyperplane, is Θ(n). 1