Disjoint path covers joining prescribed source and sink sets in interval graphs

A disjoint path cover of a graph is a set of internally vertex-disjoint paths that altogether cover every vertex of the graph. Given two disjoint source and sink sets, S and T, in a graph G, a k-disjoint path cover of G joining S and T is a disjoint path cover composed of k paths, each of which runs from a source in S to a sink in T. In this paper, we give two short proofs for the characterization of interval graphs that possess a k-disjoint path cover (of one-to-one type) joining S and T for any S and T with |S|=|T|=1 and for the characterization of interval graphs that possess a k-disjoint path cover (of one-to-many type) joining S and T for any S and T with |S|=1 and |T|=k. Also, some partial results on the many-to-many disjoint path coverability of an interval graph are addressed. In addition, we discuss interval graphs in which the disjoint-path-cover property is preserved after removal of arbitrary f or less vertices.