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- A. Grüne, R. Klein, C. Miori, S. Segura Gomis
- 2005

Let K ⊂ R be a compact convex set in the plane. A halving chord of K is a line segment pp̂, p, p̂ ∈ ∂K, which divides the area of K into two equal parts. For every direction v there exists exactly one halving chord. Its length hA(v) is the corresponding (area) halving distance. In this article we give inequalities relating the minimum and maximum (area)… (More)

- Adrian Dumitrescu, Annette Ebbers-Baumann, Ansgar Grüne, Rolf Klein, Günter Rote
- Comput. Geom.
- 2007

Let G be an embedded planar graph whose edges are curves. The detour between two points p and q (on edges or vertices) of G is the ratio between the length of a shortest path connecting p and q in G and their Euclidean distance |pq|. The maximum detour over all pairs of points is called the geometric dilation δ(G). Ebbers-Baumann, Grüne and Klein have shown… (More)

- Annette Ebbers-Baumann, Ansgar Grüne, Rolf Klein
- ISAAC
- 2003

Let G be an embedded planar graph whose edges may be curves. For two arbitrary points of G, we can compare the length of the shortest path in G connecting them against their Euclidean distance. The maximum of all these ratios is called the geometric dilation of G. Given a finite point set, we would like to know the smallest possible dilation of any graph… (More)

- Annette Ebbers-Baumann, Ansgar Grüne, Rolf Klein
- Algorithmica
- 2005

Let G be an embedded planar graph whose edges may be curves. For two arbitrary points of G, we can compare the length of the shortest path in G connecting them against their Euclidean distance. The supremum of all these ratios is called the geometric dilation of G. Given a finite point set, we would like to know the smallest possible dilation of any graph… (More)

- Florian Berger, Alexander Gilbers, Ansgar Grüne, Rolf Klein
- Algorithms
- 2009

We consider a pursuit-evasion problem where some lions have the task to clear a grid graph whose nodes are initially contaminated. The contamination spreads one step per time unit in each direction not blocked by a lion. A vertex is cleared from its contamination whenever a lion moves to it. Brass et al. [5] showed that n 2 lions are not enough to clear the… (More)

- Ansgar Grüne
- 2006

Let G be a geometric graph in the plane whose edges may be curves. For two arbitrary points on its edges, we can compare the length of the shortest path in G connecting them against their Euclidean distance. The supremum of all these ratios is called the geometric dilation of G. Given a finite point set S, we would like to know the smallest possible… (More)

- Ansgar Grüne, Sanaz Kamali
- Comput. Geom.
- 2008

Given S1, a finite set of points in the plane, we define a sequence of point sets Si as follows: With Si already determined, let Li be the set of all the line segments connecting pairs of points of ⋃i j=1 Sj , and let Si+1 be the set of intersection points of those line segments in Li, which cross but do not overlap. We show that with the exception of some… (More)

A pride of lions are prowling among the vertices and edges of an n×n grid. If their paths are known in advance, is it possible to design a safe path for a man that avoids all lions, assuming that man and lion move at the same speed? In their recent paper [4], Dumitrescu et al. employed probabilistic arguments to show that O( √ n) lions can always be… (More)

- Adrian Dumitrescu, Ansgar Grüne, Günter Rote
- EuroCG
- 2005

Let G be an embedded planar graph whose edges are curves. The detour between two points p and q (on edges or vertices) of G is the length of a shortest path connecting p and q in G divided by their Euclidean distance |pq|. The maximum detour over all pairs of points is called the geometric dilation δ(G). EbbersBaumann, Grüne and Klein have shown that every… (More)

Let G be an embedded planar graph whose edges may be curves. The detour between two points, p and q (on edges or vertices) of G, is the ratio between the shortest path in G between p and q and their Euclidean distance. The supremum over all pairs of points of all these ratios is called the geometric dilation of G. Our research is motivated by the problem of… (More)