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

In this paper,we study a variant of thepath cover problem, namely, the k -fixed-endpoint path cover problem. Given a graph G and a subset T of k vertices of V (G), a k fixed-endpoint path cover of G with respect to T is a set of vertex-disjoint paths P that covers the vertices of G such that the k vertices ofT are all endpoints of the paths inP . The k -fixed-endpoint path cover problem is to find a k -fixed-endpoint path cover ofG of minimum cardinality; note that, if T is empty, that is, k = 0, the stated problem coincides with the classical path cover problem. We show that the k -fixed-endpoint path cover problem can be solved in linear time on the class of cographs. More precisely, we first establish a lower bound on the size of a minimum k -fixed-endpoint path cover of a cograph and prove structural properties for the paths of such a path cover. Then, based on these properties, we describe an algorithm which, for a cograph G on n vertices and m edges, computes aminimum k -fixed-endpoint path cover of G in linear time, that is, in O(n + m) time. The proposedalgorithm is simple, requires linear space, andalso enables us to solve some path cover related problems, such as the 1HP and 2HP, on cographs within the same time and space complexity. © 2007 Wiley Periodicals, Inc. NETWORKS, Vol. 50(4), 231–24