Separating subgraphs in k-trees: Cables and caterpillars

The class of k-trees has the property that the minimal sets of vertices separating two nonadjacent vertices u and v of a k-tree Q induce k-complete subgraphs. We show that the union T of these subgraphs belongs to a subclass of (k - 1)-trees which generalizes caterpillars. The maximum order of a monochromatic set of vertices in the optimal coloring of this (k - 1)-tree T determines the length of the minimal collection of k vertex-disjoint paths between the two vertices of Q, the u, v-cable, which is spanned on all vertices of T.