Introduction

The Hausdorff metric $h$ imposes a geometry on the hyperspace $\mathcal{H}(\mathbb{R}^N)$ of all nonempty compact subsets of $\mathbb{R}^N$. One notion of betweenness in this geometry is an extension of betweenness in Euclidean geometry. For certain positive integers $m$, there exists a pair of disjoint finite sets $A$ and $B$ for which there are $m = \char93 ([A,B])$ different sets on the line segment defined by these sets at every distance from one of the sets. There are many fascinating and interesting properties of these numbers $m = \char93 ([A,B])$. For example, for each integer $m$ between $1$ and $18$ there exist sets $A$ and $B$ such that $\char93 ([A,B]) = m$, but no such sets exist for $m = 19$. Further discussion of this can be found in Section .

The number $\char93 ([A,B])$ is related to edge covers of bipartite graphs, which is explained in more detail in Section . For each fixed value of $m$, varying the size of one of the partite sets yields different numbers of edge covers, generating integer sequences. These sequences help us understand more about line segments in $\mathcal{H}(\mathbb{R}^N)$ and edge covers of graphs.