# Problems remaining NP-complete for sparse or dense graphs

Discussiones Mathematicae Graph Theory (1995)

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

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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

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