On a relation between $k$-path partition and $k$-path vertex cover

Abstract

The minimum vertex cover problem and the minimum vertex partition problem are central problems in graph theory and many generalisations are known. Two examples are the minimum $k$-path vertex cover problem (M$k$PVCP for short), which asks for a minimum set of vertices covering every path of order $k$, and the minimum $k$-path partition problem (M$k$PPP for short), which asks for a minimum set of paths in a maximal path packing whose every path has at most $k$ vertices.

In this paper, we will present a relation between the M$k$PPP and the M$k$PVCP. Based on it, we will obtain new bounds for their invariants and a new sufficient condition for NP-hardness of the M$k$PVCP in terms of forbidden subgraphs.