Extended skew partition problem
A skew partition as defined by Chvátal is a partition of the vertex set of a graph into four nonempty parts A 1 , A 2 , B 1 , B 2 such that there are all possible edges between A 1 and A 2 , and no edges between B 1 and B 2 . We introduce the concept of ( n 1 , n 2 ) -extended skew partition which i...
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| Veröffentlicht in: | Discrete mathematics Jg. 306; H. 19; S. 2438 - 2449 |
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| Sprache: | Englisch |
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Elsevier B.V
06.10.2006
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| ISSN: | 0012-365X, 1872-681X |
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| Abstract | A skew partition as defined by Chvátal is a partition of the vertex set of a graph into four nonempty parts
A
1
,
A
2
,
B
1
,
B
2
such that there are all possible edges between
A
1
and
A
2
, and no edges between
B
1
and
B
2
. We introduce the concept of
(
n
1
,
n
2
)
-extended skew partition which includes all partitioning problems into
n
1
+
n
2
nonempty parts
A
1
,
…
,
A
n
1
,
B
1
,
…
,
B
n
2
such that there are all possible edges between the
A
i
parts, no edges between the
B
j
parts,
i
∈
{
1
,
…
,
n
1
}
,
j
∈
{
1
,
…
,
n
2
}
, which generalizes the skew partition. We present a polynomial-time algorithm for testing whether a graph admits an
(
n
1
,
n
2
)
-extended skew partition. As a tool to complete this task we also develop a generalized 2-SAT algorithm, which by itself may have application to other partition problems. |
|---|---|
| AbstractList | A skew partition as defined by Chvátal is a partition of the vertex set of a graph into four nonempty parts
A
1
,
A
2
,
B
1
,
B
2
such that there are all possible edges between
A
1
and
A
2
, and no edges between
B
1
and
B
2
. We introduce the concept of
(
n
1
,
n
2
)
-extended skew partition which includes all partitioning problems into
n
1
+
n
2
nonempty parts
A
1
,
…
,
A
n
1
,
B
1
,
…
,
B
n
2
such that there are all possible edges between the
A
i
parts, no edges between the
B
j
parts,
i
∈
{
1
,
…
,
n
1
}
,
j
∈
{
1
,
…
,
n
2
}
, which generalizes the skew partition. We present a polynomial-time algorithm for testing whether a graph admits an
(
n
1
,
n
2
)
-extended skew partition. As a tool to complete this task we also develop a generalized 2-SAT algorithm, which by itself may have application to other partition problems. |
| Author | Gravier, Sylvain Klein, Sulamita de Figueiredo, Celina M.H. Dantas, Simone |
| Author_xml | – sequence: 1 givenname: Simone surname: Dantas fullname: Dantas, Simone organization: COPPE, Universidade Federal do Rio de Janeiro, Brazil – sequence: 2 givenname: Celina M.H. surname: de Figueiredo fullname: de Figueiredo, Celina M.H. email: celina@cos.ufrj.br organization: Instituto de Matemática and COPPE, Universidade Federal do Rio de Janeiro, Brazil – sequence: 3 givenname: Sylvain surname: Gravier fullname: Gravier, Sylvain organization: CNRS, Laboratoire Leibniz, France – sequence: 4 givenname: Sulamita surname: Klein fullname: Klein, Sulamita organization: Instituto de Matemática and COPPE, Universidade Federal do Rio de Janeiro, Brazil |
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| Cites_doi | 10.1016/S0166-218X(03)00371-8 10.1006/jctb.1997.1812 10.1016/0020-0190(79)90002-4 10.1016/0095-8956(85)90049-8 10.1007/s004939970003 10.1145/301250.301373 10.1137/S0895480100384055 10.1007/BF01204715 10.1016/0012-365X(92)90588-7 10.1006/jagm.1999.1122 |
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| Issue | 19 |
| Keywords | Algorithms and data structures Skew partition Computational and structural complexity 2-SAT |
| Language | English |
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| References | Chvátal (bib5) 1985; 39 Fleischner, Stiebitz (bib12) 1992; 101 Feder, Hell, Klein, Motwani (bib11) 2003; 16 American Institute of Mathematics, 2002. S. Dantas, C.M.H. de Figueiredo, S. Gravier, S. Klein, On H-partition problems, Relat ´orio T ´ecnico COPPE/Engenharia de Sistemas e Computação, ES-579/02, Maio—2002. P. Hell, S. Klein, L.T. Nogueira, F. Protti, Independent Alon, Tarsi (bib1) 1992; 12 Feder, Hell, Huang (bib9) 1999; 19 K. Cameron, E.M. Eschen, C.T. Hoàng, R. Sritharan, The list partition problem for graphs, in: Proceedings of SODA 2004—ACM—SIAM Symposium on Discrete Algorithms. T. Feder, P. Hell, S. Klein, R. Motwani, Complexity of graph partition problems, in: Proceedings of the 31st Annual ACM Symposium on Theory of Computing, 1999, pp. 464–472. Feder, Hell (bib8) 1998; 72 M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, Strong perfect graph theorem, in: Perfect Graph Conjecture Workshop, P. Seymour, R. Thomas, organizers Aspvall, Plass, Tarjan (bib2) 1979; 8 de Figueiredo, Klein, Kohayakawa, Reed (bib7) 2000; 37 Hell, Klein, Nogueira, Protti (bib13) 2004; 143 s in chordal graphs, in: Proceedings of CLAIO 2002—XI Latin–Iberian American Congress of Operations Research, 2002, pp. 1–11. 10.1016/j.disc.2005.12.034_bib6 10.1016/j.disc.2005.12.034_bib4 Feder (10.1016/j.disc.2005.12.034_bib11) 2003; 16 10.1016/j.disc.2005.12.034_bib3 10.1016/j.disc.2005.12.034_bib10 10.1016/j.disc.2005.12.034_bib14 Fleischner (10.1016/j.disc.2005.12.034_bib12) 1992; 101 Hell (10.1016/j.disc.2005.12.034_bib13) 2004; 143 de Figueiredo (10.1016/j.disc.2005.12.034_bib7) 2000; 37 Chvátal (10.1016/j.disc.2005.12.034_bib5) 1985; 39 Aspvall (10.1016/j.disc.2005.12.034_bib2) 1979; 8 Feder (10.1016/j.disc.2005.12.034_bib8) 1998; 72 Feder (10.1016/j.disc.2005.12.034_bib9) 1999; 19 Alon (10.1016/j.disc.2005.12.034_bib1) 1992; 12 |
| References_xml | – volume: 39 start-page: 189 year: 1985 end-page: 199 ident: bib5 article-title: Star-cutsets and perfect graphs publication-title: J. Combin. Theory Ser. B – reference: K. Cameron, E.M. Eschen, C.T. Hoàng, R. Sritharan, The list partition problem for graphs, in: Proceedings of SODA 2004—ACM—SIAM Symposium on Discrete Algorithms. – reference: S. Dantas, C.M.H. de Figueiredo, S. Gravier, S. Klein, On H-partition problems, Relat ´orio T ´ecnico COPPE/Engenharia de Sistemas e Computação, ES-579/02, Maio—2002. – reference: P. Hell, S. Klein, L.T. Nogueira, F. Protti, Independent – reference: T. Feder, P. Hell, S. Klein, R. Motwani, Complexity of graph partition problems, in: Proceedings of the 31st Annual ACM Symposium on Theory of Computing, 1999, pp. 464–472. – volume: 101 start-page: 39 year: 1992 end-page: 48 ident: bib12 article-title: A solution of a coloring problem of P. Erdős publication-title: Discrete Math. – volume: 16 start-page: 449 year: 2003 end-page: 478 ident: bib11 article-title: List partitions publication-title: SIAM J. Discrete Math. – volume: 37 start-page: 505 year: 2000 end-page: 521 ident: bib7 article-title: Finding skew partitions efficiently publication-title: J. Algorithms – reference: M. Chudnovsky, N. Robertson, P. Seymour, R. Thomas, Strong perfect graph theorem, in: Perfect Graph Conjecture Workshop, P. Seymour, R. Thomas, organizers, – reference: 's in chordal graphs, in: Proceedings of CLAIO 2002—XI Latin–Iberian American Congress of Operations Research, 2002, pp. 1–11. – volume: 19 start-page: 487 year: 1999 end-page: 505 ident: bib9 article-title: List homomorphisms and circular arc graphs publication-title: Combinatorica – volume: 72 start-page: 236 year: 1998 end-page: 250 ident: bib8 article-title: List homomorphisms to reflexive graphs publication-title: J. Combin. Theory Ser. B – volume: 12 start-page: 125 year: 1992 end-page: 134 ident: bib1 article-title: Colorings and orientations of graphs publication-title: Combinatorica – reference: , American Institute of Mathematics, 2002. – volume: 143 start-page: 185 year: 2004 end-page: 194 ident: bib13 article-title: Partitioning chordal graphs into independent sets and cliques publication-title: Discrete Appl. Math. – volume: 8 start-page: 121 year: 1979 end-page: 123 ident: bib2 article-title: A linear-time algorithm for testing the truth of certain quantified boolean formulas publication-title: Inform. Process. Lett. – volume: 143 start-page: 185 year: 2004 ident: 10.1016/j.disc.2005.12.034_bib13 article-title: Partitioning chordal graphs into independent sets and cliques publication-title: Discrete Appl. Math. doi: 10.1016/S0166-218X(03)00371-8 – ident: 10.1016/j.disc.2005.12.034_bib14 – volume: 72 start-page: 236 year: 1998 ident: 10.1016/j.disc.2005.12.034_bib8 article-title: List homomorphisms to reflexive graphs publication-title: J. Combin. Theory Ser. B doi: 10.1006/jctb.1997.1812 – volume: 8 start-page: 121 year: 1979 ident: 10.1016/j.disc.2005.12.034_bib2 article-title: A linear-time algorithm for testing the truth of certain quantified boolean formulas publication-title: Inform. Process. Lett. doi: 10.1016/0020-0190(79)90002-4 – volume: 39 start-page: 189 year: 1985 ident: 10.1016/j.disc.2005.12.034_bib5 article-title: Star-cutsets and perfect graphs publication-title: J. Combin. Theory Ser. B doi: 10.1016/0095-8956(85)90049-8 – volume: 19 start-page: 487 year: 1999 ident: 10.1016/j.disc.2005.12.034_bib9 article-title: List homomorphisms and circular arc graphs publication-title: Combinatorica doi: 10.1007/s004939970003 – ident: 10.1016/j.disc.2005.12.034_bib10 doi: 10.1145/301250.301373 – volume: 16 start-page: 449 year: 2003 ident: 10.1016/j.disc.2005.12.034_bib11 article-title: List partitions publication-title: SIAM J. Discrete Math. doi: 10.1137/S0895480100384055 – volume: 12 start-page: 125 year: 1992 ident: 10.1016/j.disc.2005.12.034_bib1 article-title: Colorings and orientations of graphs publication-title: Combinatorica doi: 10.1007/BF01204715 – volume: 101 start-page: 39 year: 1992 ident: 10.1016/j.disc.2005.12.034_bib12 article-title: A solution of a coloring problem of P. Erdős publication-title: Discrete Math. doi: 10.1016/0012-365X(92)90588-7 – ident: 10.1016/j.disc.2005.12.034_bib3 – ident: 10.1016/j.disc.2005.12.034_bib4 – ident: 10.1016/j.disc.2005.12.034_bib6 – volume: 37 start-page: 505 year: 2000 ident: 10.1016/j.disc.2005.12.034_bib7 article-title: Finding skew partitions efficiently publication-title: J. Algorithms doi: 10.1006/jagm.1999.1122 |
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| Snippet | A skew partition as defined by Chvátal is a partition of the vertex set of a graph into four nonempty parts
A
1
,
A
2
,
B
1
,
B
2
such that there are all... |
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| StartPage | 2438 |
| SubjectTerms | 2-SAT Algorithms and data structures Computational and structural complexity Skew partition |
| Title | Extended skew partition problem |
| URI | https://dx.doi.org/10.1016/j.disc.2005.12.034 |
| Volume | 306 |
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