Abstract
We define the Hamiltonian completion number of a graph G, denoted hc(G), to be the minimum number of lines that need to be added to G in order to make it Hamiltonian. The Hamiltonian completion problem asks for hc(G) and a specific Hamiltonian cycle containing hc(G) new lines. We derive an efficient algorithm for finding hc(T) for any tree T, and show that if S is the set of spanning trees of an arbitrary connected graph G, then

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A number of other general results are presented including an efficient heuristic procedure which can be used on arbitrary graphs.
Keywords
- Span Tree
- Connected Graph
- Travel Salesman Problem
- Hamiltonian Cycle
- Hamiltonian Path
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References
Karp, R. M., "Reducibility Among Combinatorial Problems," Complexity of Computer Computations (R. Miller and J. Thatcher, Eds.)
Harary, F., and Schwenk, A., "Evolution of the Path Number of a Graph, Covering and Packing in Graphs, II," Graph Theory and Computing (R.C. Read, ed.) Academic Press, New York, 1972, 39–45.
Boesch, F. T., Chen, S., and McHugh, N.A.M., "On Covering the Points of a Graph with Point Disjoint Paths," this volume p. 201.
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© 1974 Springer-Verlag Berlin
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Goodman, S., Hedetniemi, S. (1974). On the hamiltonian completion problem. In: Bari, R.A., Harary, F. (eds) Graphs and Combinatorics. Lecture Notes in Mathematics, vol 406. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066448
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DOI: https://doi.org/10.1007/BFb0066448
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-06854-9
Online ISBN: 978-3-540-37809-9
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