Partial and Perfect Path Covers of Cographs

A set P of disjoint paths in a graph G is called a (complete) path cover of G if every vertex of G belongs to one of the paths in P. A path cover of any subgraph of G is called a partial path cover of G. For xed k > 0, a k-blanket of graph G is a partial path cover P of G, consisting of exactly k paths, that maximizes the size of the subgraph covered by P. A k-core of graph G is a partial path cover P of G, consisting of exactly k paths, that minimizes the sum, over all vertices v of G, of the distance of v to its closest path in P. The problems of nding a k-blanket or a k-core (for xed k) of an arbitrary graph G as well as the dual minimum-path-cover problem ((nd a path cover of minimum size) are all NP-Hard. A linear time algorithm is known (see 4]) for the minimum-path-cover problem on cographs (graphs that can be constructed from a collection of isolated vertices by union and complement operations). However, prior to this paper, polynomial-time algorithms for the k-core problem were known only for trees-and even then for k = 1; 2 only (see 22, 2]). In this paper, we introduce a variant of a minimum path cover, called a perfect path cover. We show that every cograph has a perfect path cover, and we exploit this to obtain an O(m + n log n) time algorithm for nding, for any arbitrary k, a k-blanket or a k-core of a arbitrary cograph on n vertices and m edges.