An efficient EREW algorithm for minimum path cover and Hamiltonicity on cographs

It is shown that the notoriously difficult problem of finding the minimum number of paths that cover the vertices of a graph can be solved efficiently for cographs. The result implies that for this class of graphs finding a Hamiltonian path and a Hamiltonian cycle can be solved efficiently in parallel. Specifically, with an n-vertex cograph G represented by its parse tree as input, the algorithm determines the number of paths in a minimum path cover in O(log n) time using n/log n processors in the exclusive read exclusive write parallel random access machine (EREW-PRAM) model. The authors also exhibit all the paths in a minimum path cover of G in O(log²n) time using n/log n processors in the EREW-PRAM