Problems remaining NP-complete for sparse or dense graphs

• Autorzy: Ingo Schiermeyer
• 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

Computational Complexity; NP-Completeness; Hamiltonian Circuit; Hamiltonian Path; Independent Set

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 1995
• Tom: 15
• Numer: 1
• Strony: 33-41

Daty

wydano

1995

otrzymano

1994-04-18

poprawiono

1994-09-02

Twórcy

autor

Ingo Schiermeyer

• Lehrstuhl C für Mathematik, Technische Hochschule Aachen, D-52056 Aachen, Germany

Bibliografia

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• [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.

Identyfikatory

DOI

10.7151/dmgt.1004