A linear-time algorithm for the k-fixed-endpoint path cover problem on cographs

• Authors: Katerina Asdre;Stavros D. Nikolopoulos
• Venue: Networks
• Year: 2007

In this paper, we study a variant of the path cover problem,namely, the k-fixed-endpoint path cover problem. Given agraph G and a subset {\cal T} of k vertices ofV(G), a k-fixed-endpoint path cover ofG with respect to {\cal T} is a set of vertex-disjoint paths{\cal P} that covers the vertices of G such that thek vertices of {\cal T} are all endpoints of the paths in{\cal P}. The k-fixed-endpoint path cover problem is to finda k-fixed-endpoint path cover of G of minimumcardinality; note that, if {\cal T} is empty, that is, k =0, the stated problem coincides with the classical path coverproblem. We show that the k-fixed-endpoint path coverproblem can be solved in linear time on the class of cographs. Moreprecisely, we first establish a lower bound on the size of aminimum k-fixed-endpoint path cover of a cograph and provestructural properties for the paths of such a path cover. Then,based on these properties, we describe an algorithm which, for acograph G on n vertices and m edges, computesa minimum k-fixed-endpoint path cover of G in lineartime, that is, in O(n+m) time. The proposedalgorithm is simple, requires linear space, and also enables us tosolve some path cover related problems, such as the 1HP and 2HP, oncographs within the same time and space complexity. © 2007Wiley Periodicals, Inc. NETWORKS, Vol. 50(4), 231240 2007