Polynomial algorithms that prove an NP-hard hypothesis implies an NP-hard conclusion

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Bauer, D. and Broersma, H.J. and Morgana, A. and Schmeichel, E. (1999) Polynomial algorithms that prove an NP-hard hypothesis implies an NP-hard conclusion. [Report]

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Abstract: A number of results in Hamiltonian graph theory are of the form $\mathcal{P}$$_{1}$ implies $\mathcal{P}$$_{2}$, where $\mathcal{P}$$_{1}$ is a property of graphs that is NP-hard and $\mathcal{P}$$_{2}$ is a cycle structure property of graphs that is also NP-hard. Such a theorem is the well-known Chv\'{a}tal-Erd\"{o}s Theorem, which states that every graph $G$ with $\alpha \leq \kappa $ is Hamiltonian. Here $\kappa $ is the vertex connectivity of $G$ and $\alpha $ is the cardinality of a largest set of independent vertices of $G$. In another paper Chv\'{a}tal points out that the proof of this result is in fact a polynomial time construction that either produces a Hamilton cycle or a set of more than $\kappa $ independent vertices. In this note we point out that other theorems in Hamiltonian graph theory have a similar character. In particular, we present a constructive proof of the well-known theorem of Jung for graphs on $16$ or more vertices..
Item Type:Report
Additional information:Imported from MEMORANDA
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Electrical Engineering, Mathematics and Computer Science (EEMCS)
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Link to this item:http://purl.utwente.nl/publications/65687
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