The Hausdorff metric provides a way to measure the distance between nonempty compact sets in , from which we can build a geometry of sets. This geometry is very different than the standard Euclidean geometry and provides many interesting results. In this paper we focus on line segments in this geometry, where pairs of disjoint sets and satisfying certain distance conditions have the property that there are exactly different sets on the line segment at every distance from , where can assume many values different than one. We provide new families of sets that generate previously unrecorded integer sequences via these values of by connecting the values of to the number of edge coverings of a graph corresponding to the sets and .