On the weightedk-path vertex cover on interval graphs

We consider a version of the k-path vertex cover problem that asks for the minimum weight subset C of vertices of a graph G such that every path on k vertices in G has at least one vertex in common with C. We present two dynamic algorithms solving this problem on interval graphs. The first one works on general interval graphs but is in practice limited to small values of k.
The second algorithm computes minimum weight vertex cover for arbitrary k on proper interval graph G = (V,E) in time O(|V|^2|E|).