# Problems remaining NP-complete for sparse or dense graphs

Discussiones Mathematicae Graph Theory (1995)

- Volume: 15, Issue: 1, page 33-41
- ISSN: 2083-5892

## Access Full Article

top## Abstract

top## How to cite

topIngo Schiermeyer. "Problems remaining NP-complete for sparse or dense graphs." Discussiones Mathematicae Graph Theory 15.1 (1995): 33-41. <http://eudml.org/doc/270554>.

@article{IngoSchiermeyer1995,

abstract = {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’.},

author = {Ingo Schiermeyer},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {Computational Complexity; NP-Completeness; Hamiltonian Circuit; Hamiltonian Path; Independent Set; computational complexity; NP-completeness; Hamiltonian path; independent set; NP-complete; Hamiltonian circuit},

language = {eng},

number = {1},

pages = {33-41},

title = {Problems remaining NP-complete for sparse or dense graphs},

url = {http://eudml.org/doc/270554},

volume = {15},

year = {1995},

}

TY - JOUR

AU - Ingo Schiermeyer

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

JO - Discussiones Mathematicae Graph Theory

PY - 1995

VL - 15

IS - 1

SP - 33

EP - 41

AB - 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’.

LA - eng

KW - Computational Complexity; NP-Completeness; Hamiltonian Circuit; Hamiltonian Path; Independent Set; computational complexity; NP-completeness; Hamiltonian path; independent set; NP-complete; Hamiltonian circuit

UR - http://eudml.org/doc/270554

ER -

## References

top- [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). Zbl1226.05083
- [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. Zbl0789.05058
- [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. Zbl0047.17001
- [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. Zbl0746.05038
- [6] M. R. Garey and D. S. Johnson, Computers and I, A Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979). Zbl0411.68039
- [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. Zbl0767.68057
- [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. Zbl0803.68052