Degree constrained 2-partitions of semicomplete digraphs
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| Title: | Degree constrained 2-partitions of semicomplete digraphs |
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| Authors: | Bang-Jensen, Jørgen, Christiansen, Tilde My |
| Source: | Bang-Jensen, J & Christiansen, T M 2018, ' Degree constrained 2-partitions of semicomplete digraphs ', Theoretical Computer Science, vol. 746, pp. 112-123 . https://doi.org/10.1016/j.tcs.2018.06.028 |
| Publisher Information: | Elsevier BV, 2018. |
| Publication Year: | 2018 |
| Subject Terms: | Semicomplete digraph, Minimum out-degree, 2-partition, Digraphs of bounded independence number, 0211 other engineering and technologies, 0102 computer and information sciences, 02 engineering and technology, Minimum in-degree, Tournament, 01 natural sciences, Minimum semi-degree, NP-complete |
| Description: | A 2-partition of a digraph D is a partition ( V 1 , V 2 ) of V ( D ) into two disjoint non-empty sets V 1 and V 2 such that V 1 ∪ V 2 = V ( D ) . A semicomplete digraph is a digraph with no pair of non-adjacent vertices. We consider the complexity of deciding whether a given semicomplete digraph has a 2-partition such that each part of the partition induces a (semicomplete) digraph with some specified property. In [4] and [5] Bang-Jensen, Cohen and Havet determined the complexity of 120 such 2-partition problems for general digraphs. Several of these problems are NP-complete for general digraphs and thus it is natural to ask whether this is still the case for well-structured classes of digraphs, such as semicomplete digraphs. This is the main topic of the paper. More specifically, we consider 2-partition problems where the set of properties are minimum out-, minimum in- or minimum semi-degree. Among other results we prove the following: • For all integers k 1 , k 2 ≥ 1 and k 1 + k 2 ≥ 3 it is NP-complete to decide whether a given digraph D has a 2-partition ( V 1 , V 2 ) such that D 〈 V i 〉 has out-degree at least k i for i = 1 , 2 . • For every fixed choice of integers α , k 1 , k 2 ≥ 1 there exists a polynomial algorithm for deciding whether a given digraph of independence number at most α has a 2-partition ( V 1 , V 2 ) such that D 〈 V i 〉 has out-degree at least k i for i = 1 , 2 . • For every fixed integer k ≥ 1 there exists a polynomial algorithm for deciding whether a given semicomplete digraph has a 2-partition ( V 1 , V 2 ) such that D 〈 V 1 〉 has out-degree at least one and D 〈 V 2 〉 has in-degree at least k. • It is NP-complete to decide whether a given semicomplete digraph D has a 2-partition ( V 1 , V 2 ) such that D 〈 V i 〉 is a strong tournament. |
| Document Type: | Article |
| File Description: | application/pdf |
| Language: | English |
| ISSN: | 0304-3975 |
| DOI: | 10.1016/j.tcs.2018.06.028 |
| Access URL: | https://findresearcher.sdu.dk:8443/ws/files/143341866/Degree_constrained_2_partitions_of_semicomplete_digraphs.pdf https://dblp.uni-trier.de/db/journals/tcs/tcs746.html#Bang-JensenC18 https://www.sciencedirect.com/science/article/abs/pii/S0304397518304444 |
| Rights: | Elsevier Non-Commercial CC BY NC ND |
| Accession Number: | edsair.doi.dedup.....f633ef010ec77de60dbb6da85aef2133 |
| Database: | OpenAIRE |
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Nájsť tento článok vo Web of Science | Abstract: | A 2-partition of a digraph D is a partition ( V 1 , V 2 ) of V ( D ) into two disjoint non-empty sets V 1 and V 2 such that V 1 ∪ V 2 = V ( D ) . A semicomplete digraph is a digraph with no pair of non-adjacent vertices. We consider the complexity of deciding whether a given semicomplete digraph has a 2-partition such that each part of the partition induces a (semicomplete) digraph with some specified property. In [4] and [5] Bang-Jensen, Cohen and Havet determined the complexity of 120 such 2-partition problems for general digraphs. Several of these problems are NP-complete for general digraphs and thus it is natural to ask whether this is still the case for well-structured classes of digraphs, such as semicomplete digraphs. This is the main topic of the paper. More specifically, we consider 2-partition problems where the set of properties are minimum out-, minimum in- or minimum semi-degree. Among other results we prove the following: • For all integers k 1 , k 2 ≥ 1 and k 1 + k 2 ≥ 3 it is NP-complete to decide whether a given digraph D has a 2-partition ( V 1 , V 2 ) such that D 〈 V i 〉 has out-degree at least k i for i = 1 , 2 . • For every fixed choice of integers α , k 1 , k 2 ≥ 1 there exists a polynomial algorithm for deciding whether a given digraph of independence number at most α has a 2-partition ( V 1 , V 2 ) such that D 〈 V i 〉 has out-degree at least k i for i = 1 , 2 . • For every fixed integer k ≥ 1 there exists a polynomial algorithm for deciding whether a given semicomplete digraph has a 2-partition ( V 1 , V 2 ) such that D 〈 V 1 〉 has out-degree at least one and D 〈 V 2 〉 has in-degree at least k. • It is NP-complete to decide whether a given semicomplete digraph D has a 2-partition ( V 1 , V 2 ) such that D 〈 V i 〉 is a strong tournament. |
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| ISSN: | 03043975 |
| DOI: | 10.1016/j.tcs.2018.06.028 |