The number of edge covers of bipartite graphs or of shortest paths with fixed endpoints in the space of compact sets in Rn

Abstract

The possible number of shortest paths joining points in the metric space of compact sets in Euclidean space endowed with the Hausdorff metric is studied. For all n ≤ 1000, except eight values, it is checked whether n can equal the number of such shortest paths. In particular, new lacunas are found, namely 41, 59, and 67 (previously, only two such lacunas, 19 and 37, were known).