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

Abstract

AbstractIn this paper, we study a variant of the path cover problem, namely, the k‐fixed‐endpoint path cover problem. Given a graph G and a subset ${\cal T}$ of k vertices of V(G), a k‐fixed‐endpoint path cover of G with respect to ${\cal T}$ is a set of vertex‐disjoint paths ${\cal P}$ that covers the vertices of G such that the k vertices of ${\cal T}$ are all endpoints of the paths in ${\cal P}$. The k‐fixed‐endpoint path cover problem is to find a k‐fixed‐endpoint path cover of G of minimum cardinality; note that, if ${\cal T}$ 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 a minimum k‐fixed‐endpoint path cover of G in linear time, that is, in O(n+m) time. The proposed algorithm is simple, requires linear space, and also 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), 231–240 2007