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In this paper, we consider the problem of curve reconstruction from a finite planar set of points. To solve this problem, we propose to use a family of neighborhood graphs included in the Gabriel graph. The neighborhood that we use is the beta-neighborhood, initially defined in the context of circle-based beta-skeletons, but applied to edges of the Voronoi diagram. This family of graphs includes the local crust. This formulation enables us to design effective algorithms to reconstruct curves, by using as a prior knowledge that the curves to be reconstructed are without intersections. We show, through several examples, that the proposed algorithms improve the results obtained with the local crust, when the set of points is of low density.