On the k-path cover problem for cacti

In this paper we investigate the k-path cover problem for graphs, which is to find the minimum number of vertex disjoint k-paths that cover all the vertices of a graph. The k-path cover problem for general graphs is NP-complete. Though notable applications of this problem to database design, network, VLSI design, ring protocols, and code optimization, efficient algorithms are known for only few special classes of graphs. In order to solve this problem for cacti, i.e., graphs where no edge lies on more than one cycle, we introduce the so-called Steiner version of the k-path cover problem, and develop an efficient algorithm for the Steiner k-path cover problem for cacti, which finds an optimal k-path cover for a given cactus in polynomial time.