Abstract
Nakano et al. [A time-optimal solution for the path cover problem on cographs, Theoret. Comput. Science 290 (2003) 1541–1556] presented a time- and work-optimal algorithm for finding the smallest number of vertex-disjoint paths that cover the vertices of a cograph and left open the problem of applying their technique into other classes of graphs. Motivated by this issue we generalize their technique and apply it to the class of P 4 -sparse graphs, which forms a proper superclass of cographs. We show that the path cover problem on P 4 -sparse graphs can also be optimally solved. More precisely, given a P 4 -sparse graph G on n vertices and its modular decomposition tree, we describe an optimal parallel algorithm which returns a minimum path cover of G in O ( log n ) time using O ( n / log n ) processors on the EREW PRAM model. Our results generalize previous results and extend the family of perfect graphs admitting optimal solutions for the path cover problem.
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