On the geometric dilation of curves and point sets
نویسندگان
چکیده
Let G be an embedded planar graph whose edges are curves. The detour between two points u and v (on edges or vertices) of G is the ratio between the shortest path in G between u and v and their Euclidean distance. The maximum detour over all pairs of points is called the geometric dilation δ(G). Ebbers-Baumann, Grüne and Klein have recently shown that every finite point set is contained in a planar graph whose geometric dilation is at most 1.678, and some point sets require graphs with dilation δ ≥ π/2 ≈ 1.57. We prove a stronger lower bound δ ≥ (1 + 10)π/2 by relating graphs with small dilation to a problem of packing and covering the plane by circular disks.
منابع مشابه
ar X iv : m at h . M G / 0 40 71 35 v 2 2 5 A ug 2 00 5 On the Geometric Dilation of Closed Curves , Graphs , and Point Sets ∗
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 that every finite point set is contai...
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