β-skeletons for a set of line segments in R²

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β-skeletons are well-known neighborhood graphs for a set of points. We extend this notion to sets of line segments in the Euclidean plane and present algorithms computing such skeletons for the entire range of β values. The main reason of such extension is the possibility to study β-skeletons for points moving along given line segments. We show that relations between β-skeletons for β> 1, 1-skeleton (Gabriel Graph), and the Delaunay triangulation for sets of points hold also for sets of segments. We present algorithms for computing circle-based and lune-based β-skeletons. We describe an algorithm that for β≥ 1 computes the β-skeleton for a set S of n segments in the Euclidean plane in O(n² α(n) n) time in the circle-based case and in O(n² λ₄(n)) in the lune-based one, where the construction relies on the Delaunay triangulation for S, α is a functional inverse of Ackermann function and λ₄(n) denotes the maximum possible length of a (n,4) Davenport-Schinzel sequence. When 0 < β< 1, the β-skeleton can be constructed in a O(n³ λ₄(n)) time. In the special case of β= 1, which is a generalization of Gabriel Graph, the construction can be carried out in a O(n n) time.