# On the Spanning Ratio of Gabriel Graphs and β-skeletons

## Abstract

The spanning ratio of a graph defined on *n* points in the Euclidean plane is the maximal ratio over all pairs of data points (*u*, *v*), of the minimum graph distance between *u* and *v*, over the Euclidean distance between *u* and *v*. A connected graph is said to be a *k*-spanner if the spanning ratio does not exceed *k*. For example, for any *k*, there exists a point set whose minimum spanning tree is not a *k*-spanner. At the other end of the spectrum, a Delaunay triangulation is guaranteed to be a 2.42- spanner[11]. For proximity graphs *inbetween* these two extremes, such as Gabriel graphs[8], relative neighborhood graphs[16] and β-skeletons[12] with β ∈ [0, 2] some interesting questions arise. We show that the spanning ratio for Gabriel graphs (which are β-skeletons with β = 1) is Θ(√*n*) in the worst case. For all β-skeletons with β ∈ [0, 1], we prove that the spanning ratio is at most *O*(*n**λ*) where λ = (1 - log_{2}(1 +√1 - β^{2}))/2. For all β-skeletons with β ∈ [1, 2), we prove that there exist point sets whose spanning ratio is at least (^{1}_{2} - *o*(1))√*n*. For relative neighborhood graphs[16] (skeletons with β = 2), we show that there exist point sets where the spanning ratio is ω(*n*). For points drawn independently from the uniform distribution on the unit square, we show that the spanning ratio of the (random) Gabriel graph and all β-skeletons with β ∈ [1, 2] tends to ∞ in probability as √log *n*/ log log *n*.

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