Ansgar Grüne

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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)
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)
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)
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)
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)
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)