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Optimal Hamiltonian completions and path covers for trees, and a reduction to maximum flow

  • D. S. Franzblau (a1) and A. Raychaudhuri (a2)
Abstract

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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[2]F. T. Boesch and J. F. Gimpel , “Covering the points of a digraph with point-disjoint paths and its application to code optimization”, JACM 24 (1977) 192198.

[3]J. A. Bondy and U. S. R. Murty , Graph theory with applications (North Holland, NY, 1976).

[5]L. R. Ford and D. R Fulkerson , “A simple algorithm for finding maximal network flows and an application to the Hitchcock problem”, Canadian J. Math. 9 (1957) 210218.

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[7]S. E. Goodman , S. T. Hedetniemi and P. J. Slater , “Advances on the Hamiltonian completion problem”, JACM. 22 (1975) 352360.

[11]S. Kundu , “A linear algorithm for the Hamiltonian completion number of a tree”, Info. Proc. Letters 5 (1976) 5557.

[13]S. Moran and Y. Wolfstahl , “Optimal covering of cacti by vertex-disjoint paths”, Theoret. Comp. Sci. 84 (1991) 179197.

[14]S. S. Pinter and Y. Wolfstahl , “On mapping processes to processors in distributed systems”, Internat. J. Parallel Prog. 16 (1987) 115.

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The ANZIAM Journal
  • ISSN: 1446-1811
  • EISSN: 1446-8735
  • URL: /core/journals/anziam-journal
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