For each fixed pair α,c > 0 let INDEPENDENT SET () and INDEPENDENT SET () be the problem INDEPENDENT SET restricted to graphs on n vertices with or edges, respectively. Analogously, HAMILTONIAN CIRCUIT () and HAMILTONIAN PATH () are the problems HAMILTONIAN CIRCUIT and HAMILTONIAN PATH restricted to graphs with 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’.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1004,
author = {Ingo Schiermeyer},
title = {Problems remaining NP-complete for sparse or dense graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {15},
year = {1995},
pages = {33-41},
zbl = {0829.05042},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1004}
}
Ingo Schiermeyer. Problems remaining NP-complete for sparse or dense graphs. Discussiones Mathematicae Graph Theory, Tome 15 (1995) pp. 33-41. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1004/
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