Paths with no Small Angles

Bárány, Imre; Pór, Attila; Valtr, Pável

doi:10.1007/978-3-540-78773-0_56

Cited by 2 publications

(1 citation statement)

References 3 publications

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“…Aggarwal et al [2] prove that finding a spanning cycle for a point set which has minimal total angle cost is NP-hard, where the angle cost is defined as the sum of direction changes at the points. Regarding spanning paths, it has been conjectured that each planar point set admits a spanning path with minimum angle at least π 6 [12]; recently, a lower bound of π 9 has been presented [7].…”

Section: Related Workmentioning

confidence: 99%

Maximizing Maximal Angles for Plane Straight-Line Graphs

Aichholzer

Hackl

Hoffmann

et al.

Lecture Notes in Computer Science

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Let $G=(S, E)$ be a plane straight-line graph on a finite point set $S\subset\R^2$ in general position. The incident angles of a vertex $p \in S$ of $G$ are the angles between any two edges of $G$ that appear consecutively in the circular order of the edges incident to $p$. A plane straight-line graph is called $\phi$-open if each vertex has an incident angle of size at least $\phi$. In this paper we study the following type of question: What is the maximum angle $\phi$ such that for any finite set $S\subset\R^2$ of points in general position we can find a graph from a certain class of graphs on $S$ that is $\phi$-open? In particular, we consider the classes of triangulations, spanning trees, and paths on $S$ and give tight bounds in most cases.Comment: 15 pages, 14 figures. Apart of minor corrections, some proofs that were omitted in the previous version are now include

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Section: Related Workmentioning

confidence: 99%

Maximizing Maximal Angles for Plane Straight-Line Graphs

Aichholzer

Hackl

Hoffmann

et al.

Lecture Notes in Computer Science

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Drawing Hamiltonian Cycles with No Large Angles

2010

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Let n ≥ 4 be even. It is shown that every set S of n points in the plane can be connected by a (possibly self-intersecting) spanning tour (Hamiltonian cycle) consisting of n straight line edges such that the angle between any two consecutive edges is at most 2π/3. For n = 4 and 6, this statement is tight. It is also shown that every even-element point set S can be partitioned into at most two subsets, S 1 and S 2 , each admitting a spanning tour with no angle larger than π/2. Fekete and Woeginger conjectured that for sufficiently large even n, every n-element set admits such a spanning tour. We confirm this conjecture for point sets in convex position. A much stronger result holds for large point sets randomly and uniformly selected from an open region bounded by finitely many rectifiable curves: for any ε > 0, these sets almost surely admit a spanning tour with no angle larger than ε.

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Paths with no Small Angles

Cited by 2 publications

References 3 publications

Maximizing Maximal Angles for Plane Straight-Line Graphs

Maximizing Maximal Angles for Plane Straight-Line Graphs

Drawing Hamiltonian Cycles with No Large Angles

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