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\vskip 1cm{\LARGE\bf 
Integer Sequences from Configurations \\
in the Hausdorff Metric Geometry \\
\vskip .09in
via Edge Covers of Bipartite Graphs
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\begin{minipage}{2.8in}
\begin{center}
Steven Schlicker\\
Grand Valley State University\\
Department of Mathematics\\
1 Campus Drive\\
Allendale, MI 49401\\
USA\\
\href{mailto:schlicks@gvsu.edu}{\tt schlicks@gvsu.edu}
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\quad 
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Roman Vasquez\\
Auburn University\\
Department of Mathematics and Statistics\\
221 Parker Hall\\
Auburn, AL 36849\\
USA\\
\href{mailto:roman.vasquez@auburn.edu}{\tt roman.vasquez@auburn.edu}
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\ \\
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Rachel Wofford\\
Pacific Northwest National Laboratory\\
1100 Dexter Ave N.\\
Seattle, WA 98109\\
USA\\
\href{mailto:re.wofford@pnnl.gov}{\tt re.wofford@pnnl.gov}
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\begin{abstract} The Hausdorff metric provides a way to measure the
distance between nonempty compact sets in $\mathbb{R}^N$, 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 $A$ and $B$ satisfying certain distance conditions
have the property that there are exactly $m$ different sets on the
line segment $\overline{AB}$ at every distance from $A$, where $m$ can
assume many values different than one. We provide new families of sets
that generate previously unrecorded integer sequences via these values
of $m$ by connecting the values of $m$ to the number of edge coverings
of a graph corresponding to the sets $A$ and $B$.
\end{abstract}

\section{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 $m = \#([A,B])$ 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 $m = \#([A,B])$. For example, for each integer $m$ between $1$ and $18$ there exist sets $A$ and $B$ such that $\#([A,B]) = m$, but no such sets exist for $m = 19$. Further discussion of this can be found in Section~\ref{section:betweeness}.

The number $\#([A,B])$ is related to edge covers of bipartite graphs, which is explained in more detail in Section~\ref{section:graphs}. 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. 




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