Optimal Hamiltonian Completions and Path Covers for Trees, and a Reduction to Maximum Flow

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 lineartime 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. 1. Definitions and results A Hamiltonian completion of a finite graph G = .V ; E/ is a set of edges that, when added to E , ensure that G has a Hamiltonian path. The Hamiltonian completion number hc.G/ is the minimum number of edges required in a Hamiltonian completion for G. A Hamiltonian completion with the minimum number of edges is a minimum or optimal Hamiltonian completion. A closely related concept is that of a path cover for a graph G, which is a set of vertex-disjoint simple paths that together contain all the vertices of G. The path cover number pc.G/ is the minimum number of paths in 1Department of Mathematics, CUNY, College of Staten Island, 2800 Victory Blvd, Staten Island, NY 10314, USA; e-mail: [email protected]. 2Department of Mathematics, CUNY, College of Staten Island, 2800 Victory Blvd, Staten Island, NY 10314, USA; e-mail: [email protected]. c © Australian Mathematical Society 2002, Serial-fee code 1446-8735/02 3See [3] or [4] for graph terms not defined here.