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## Discussiones Mathematicae Graph Theory

1995 | 15 | 1 | 33-41
Tytuł artykułu

### Problems remaining NP-complete for sparse or dense graphs

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Treść / Zawartość
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Języki publikacji
EN
Abstrakty
EN
For each fixed pair α,c > 0 let INDEPENDENT SET ($m ≤ cn^α$) and INDEPENDENT SET ($m ≥ (ⁿ₂) - cn^α$) be the problem INDEPENDENT SET restricted to graphs on n vertices with $m ≤ cn^α$ or $m ≥ (ⁿ₂) - cn^α$ edges, respectively. Analogously, HAMILTONIAN CIRCUIT ($m ≤ n + cn^α$) and HAMILTONIAN PATH ($m ≤ n + cn^α$) are the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH restricted to graphs with $m ≤ n + cn^α$ edges. For each ϵ > 0 let HAMILTONIAN CIRCUIT (m ≥ (1 - ϵ)(ⁿ₂)) and HAMILTONIAN PATH (m ≥ (1 - ϵ)(ⁿ₂)) be the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH restricted to graphs with m ≥ (1 - ϵ)(ⁿ₂) edges.
We prove that these six restricted problems remain NP-complete. Finally, we consider sufficient conditions for a graph to have a Hamiltonian circuit. These conditions are based on degree sums and neighborhood unions of independent vertices, respectively. Lowering the required bounds the problem HAMILTONIAN CIRCUIT jumps from 'easy' to 'NP-complete'.
Słowa kluczowe
EN
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
33-41
Opis fizyczny
Daty
wydano
1995
otrzymano
1994-04-18
poprawiono
1994-09-02
Twórcy
autor
• Lehrstuhl C für Mathematik, Technische Hochschule Aachen, D-52056 Aachen, Germany
Bibliografia
• [1] J. A. Bondy, Longest paths and cycles in graphs of high degree, Research Report CORR 80-16, Dept. of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada (1980).
• [2] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications (Macmillan, London and Elsevier, New York, 1976).
• [3] H. J. Broersma, J. van den Heuvel and H. J. Veldman, A generalization of Ore's Theorem involving neighborhood unions, Discrete Math. 122 (1993) 37-49, doi: 10.1016/0012-365X(93)90285-2.
• [4] G. A. Dirac, Some theorems on abstract graphs, Proc. London Math. Soc. 2 (1952) 69-81, doi: 10.1112/plms/s3-2.1.69.
• [5] E. Flandrin, H. A. Jung and H. Li, Hamiltonism, degree sum and neighborhood intersections, Discrete Math. 90 (1991) 41-52, doi: 10.1016/0012-365X(91)90094-I.
• [6] M. R. Garey and D. S. Johnson, Computers and I, A Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979).
• [7] J. Kratochvíl, P. Savický and Z. Tuza, One more occurrence of variables makes jump from trivial to NP-complete, SIAM J. Comput. 22 (1993) 203-210, doi: 10.1137/0222015.
• [8] O. Ore, Note on Hamiltonian circuits, Amer. Math. Monthly 67 (1960) 55, doi: 10.2307/2308928.
• [9] I. Schiermeyer, The k-Satisfiability problem remains NP-complete for dense families, Discrete Math. 125 (1994) 343-346, doi: 10.1016/0012-365X(94)90175-9.
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