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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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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
 In Proc. 20th ACM Symp. on Computational Geometry (SoCG ’04
, 2004
"... Let P be a set of n points inÊ3, 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 − 7 different directions if n is even and at least 2n − 5 if n is odd. The bound for odd n is sharp. 1 ..."
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Let P be a set of n points inÊ3, 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 − 7 different directions if n is even and at least 2n − 5 if n is odd. The bound for odd n is sharp. 1
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.
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. We re ..."
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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. We review results providing bounds on b(n) and mention some additional observations. Let P be an npoint set in the plane (or, more generally, in R d). We define a visibilityblocking set for P as a set Q that is disjoint from P and such that every segment with endpoints in P contains at least one point of Q. If the points of P are all collinear, then there is a visibility blocking set with n − 1 points. The question raised in this note is, what is the smallest possible size of a visibilityblocking set for P having no three points collinear? That is, we let b(P): = min{Q  : Q a visibilityblocking set for P} b(n): = min{b(P) : P ⊂ R 2 with no three points collinear, P  = n}, and we would like to estimate the asymptotics of b(n) for large n.