A minimum Hamiltonian completion of a graph G is a minimum-size set of edges that, when added to G, guarantee a Hamiltonian path. Finding a Hamiltonian completion has applications to frequency assignment as well as distributed computing. If the new edges are deleted from the Hamiltonian path, one is left with a minimum path cover, a minimum-size set of vertex-disjoint paths that cover the vertices of G. For arbitrary graphs, constructing a minimum Hamiltonian completion or path cover is clearly NP-hard, but there exists a linear-time algorithm for trees. In this paper we first give a description and proof of correctness for this linear-time algorithm that is simpler and more intuitive than those given previously. We show that the algorithm extends also to unicyclic graphs. We then give a new method for finding an optimal path cover or Hamiltonian completion for a tree that uses a reduction to a maximum flow problem. In addition, we show how to extend the reduction to construct, if possible, a covering of the vertices of a bipartite graph with vertex-disjoint cycles, that is, a 2-factor.
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