The maximum 3-star packing problem in claw-free cubic graphs
A 3-star is a complete bipartite graph K 1 , 3 . A 3-star packing of a graph G is a collection of vertex-disjoint subgraphs of G in which each subgraph is a 3-star. The maximum 3-star packing problem is to find a 3-star packing of a given graph with the maximum number of 3-stars. A 2 -independent se...
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| Veröffentlicht in: | Journal of combinatorial optimization 2024-07, Vol.47 (5), Article 73 |
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| container_title | Journal of combinatorial optimization |
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| creator | Xi, Wenying Lin, Wensong |
| description | A 3-star is a complete bipartite graph
K
1
,
3
. A 3-star packing of a graph
G
is a collection of vertex-disjoint subgraphs of
G
in which each subgraph is a 3-star. The maximum 3-star packing problem is to find a 3-star packing of a given graph with the maximum number of 3-stars. A 2
-independent set
of a graph
G
is a subset
S
of
V
(
G
) such that for each pair of vertices
u
,
v
∈
S
, paths between
u
and
v
are all of length at least 3. In cubic graphs, the maximum 3-star packing problem is equivalent to the maximum 2-independent set problem. The maximum 2-independent set problem was proved to be NP-hard on cubic graphs (Kong and Zhao in Congressus Numerantium 143:65–80, 2000), and the best approximation algorithm of maximum 2-independent set problem for cubic graphs has approximation ratio
8
15
(Miyano et al. in WALCOM 2017, Proceedings, pp 228–240). In this paper, we first prove that the maximum 3-star packing problem is NP-hard in claw-free cubic graphs and then design a linear-time algorithm which can find a 3-star packing of a connected claw-free cubic graph
G
covering at least
3
v
(
G
)
-
8
4
vertices, where
v
(
G
) denotes the number of vertices of
G
. |
| doi_str_mv | 10.1007/s10878-024-01115-z |
| format | Article |
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K
1
,
3
. A 3-star packing of a graph
G
is a collection of vertex-disjoint subgraphs of
G
in which each subgraph is a 3-star. The maximum 3-star packing problem is to find a 3-star packing of a given graph with the maximum number of 3-stars. A 2
-independent set
of a graph
G
is a subset
S
of
V
(
G
) such that for each pair of vertices
u
,
v
∈
S
, paths between
u
and
v
are all of length at least 3. In cubic graphs, the maximum 3-star packing problem is equivalent to the maximum 2-independent set problem. The maximum 2-independent set problem was proved to be NP-hard on cubic graphs (Kong and Zhao in Congressus Numerantium 143:65–80, 2000), and the best approximation algorithm of maximum 2-independent set problem for cubic graphs has approximation ratio
8
15
(Miyano et al. in WALCOM 2017, Proceedings, pp 228–240). In this paper, we first prove that the maximum 3-star packing problem is NP-hard in claw-free cubic graphs and then design a linear-time algorithm which can find a 3-star packing of a connected claw-free cubic graph
G
covering at least
3
v
(
G
)
-
8
4
vertices, where
v
(
G
) denotes the number of vertices of
G
.</description><identifier>ISSN: 1382-6905</identifier><identifier>EISSN: 1573-2886</identifier><identifier>DOI: 10.1007/s10878-024-01115-z</identifier><language>eng</language><publisher>New York: Springer US</publisher><subject>Algorithms ; Apexes ; Approximation ; Combinatorics ; Convex and Discrete Geometry ; Graph theory ; Graphs ; Mathematical Modeling and Industrial Mathematics ; Mathematics ; Mathematics and Statistics ; Operations Research/Decision Theory ; Optimization ; Packing problem ; Theory of Computation</subject><ispartof>Journal of combinatorial optimization, 2024-07, Vol.47 (5), Article 73</ispartof><rights>The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c270t-c06e1621b602097d1918f0b83fecb111ea7e34f5eddac585bbc92524ee29de353</cites><orcidid>0000-0002-4112-0469</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10878-024-01115-z$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10878-024-01115-z$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,778,782,27907,27908,41471,42540,51302</link.rule.ids></links><search><creatorcontrib>Xi, Wenying</creatorcontrib><creatorcontrib>Lin, Wensong</creatorcontrib><title>The maximum 3-star packing problem in claw-free cubic graphs</title><title>Journal of combinatorial optimization</title><addtitle>J Comb Optim</addtitle><description>A 3-star is a complete bipartite graph
K
1
,
3
. A 3-star packing of a graph
G
is a collection of vertex-disjoint subgraphs of
G
in which each subgraph is a 3-star. The maximum 3-star packing problem is to find a 3-star packing of a given graph with the maximum number of 3-stars. A 2
-independent set
of a graph
G
is a subset
S
of
V
(
G
) such that for each pair of vertices
u
,
v
∈
S
, paths between
u
and
v
are all of length at least 3. In cubic graphs, the maximum 3-star packing problem is equivalent to the maximum 2-independent set problem. The maximum 2-independent set problem was proved to be NP-hard on cubic graphs (Kong and Zhao in Congressus Numerantium 143:65–80, 2000), and the best approximation algorithm of maximum 2-independent set problem for cubic graphs has approximation ratio
8
15
(Miyano et al. in WALCOM 2017, Proceedings, pp 228–240). In this paper, we first prove that the maximum 3-star packing problem is NP-hard in claw-free cubic graphs and then design a linear-time algorithm which can find a 3-star packing of a connected claw-free cubic graph
G
covering at least
3
v
(
G
)
-
8
4
vertices, where
v
(
G
) denotes the number of vertices of
G
.</description><subject>Algorithms</subject><subject>Apexes</subject><subject>Approximation</subject><subject>Combinatorics</subject><subject>Convex and Discrete Geometry</subject><subject>Graph theory</subject><subject>Graphs</subject><subject>Mathematical Modeling and Industrial Mathematics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Operations Research/Decision Theory</subject><subject>Optimization</subject><subject>Packing problem</subject><subject>Theory of Computation</subject><issn>1382-6905</issn><issn>1573-2886</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kDFPwzAUhC0EEqXwB5gsMRue7dhxJBZUQUGqxFJmy3Fe2pQmDXYjoL8eQ5DYmN4Nd_dOHyGXHK45QH4TOZjcMBAZA865YocjMuEql0wYo4-TlkYwXYA6JWcxbgAg6WxCbpdrpK37aNqhpZLFvQu0d_616Va0D7tyiy1tOuq37p3VAZH6oWw8XQXXr-M5OandNuLF752Sl4f75eyRLZ7nT7O7BfMihz3zoJFrwUsNAoq84gU3NZRG1ujLtBZdjjKrFVaV88qosvSFUCJDFEWFUskpuRp706K3AePebnZD6NJLK0HpTBsl8uQSo8uHXYwBa9uHpnXh03Kw35TsSMkmSvaHkj2kkBxDMZm7FYa_6n9SX5VAaeE</recordid><startdate>20240701</startdate><enddate>20240701</enddate><creator>Xi, Wenying</creator><creator>Lin, Wensong</creator><general>Springer US</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-4112-0469</orcidid></search><sort><creationdate>20240701</creationdate><title>The maximum 3-star packing problem in claw-free cubic graphs</title><author>Xi, Wenying ; Lin, Wensong</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-c06e1621b602097d1918f0b83fecb111ea7e34f5eddac585bbc92524ee29de353</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Algorithms</topic><topic>Apexes</topic><topic>Approximation</topic><topic>Combinatorics</topic><topic>Convex and Discrete Geometry</topic><topic>Graph theory</topic><topic>Graphs</topic><topic>Mathematical Modeling and Industrial Mathematics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Operations Research/Decision Theory</topic><topic>Optimization</topic><topic>Packing problem</topic><topic>Theory of Computation</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Xi, Wenying</creatorcontrib><creatorcontrib>Lin, Wensong</creatorcontrib><collection>CrossRef</collection><jtitle>Journal of combinatorial optimization</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Xi, Wenying</au><au>Lin, Wensong</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The maximum 3-star packing problem in claw-free cubic graphs</atitle><jtitle>Journal of combinatorial optimization</jtitle><stitle>J Comb Optim</stitle><date>2024-07-01</date><risdate>2024</risdate><volume>47</volume><issue>5</issue><artnum>73</artnum><issn>1382-6905</issn><eissn>1573-2886</eissn><abstract>A 3-star is a complete bipartite graph
K
1
,
3
. A 3-star packing of a graph
G
is a collection of vertex-disjoint subgraphs of
G
in which each subgraph is a 3-star. The maximum 3-star packing problem is to find a 3-star packing of a given graph with the maximum number of 3-stars. A 2
-independent set
of a graph
G
is a subset
S
of
V
(
G
) such that for each pair of vertices
u
,
v
∈
S
, paths between
u
and
v
are all of length at least 3. In cubic graphs, the maximum 3-star packing problem is equivalent to the maximum 2-independent set problem. The maximum 2-independent set problem was proved to be NP-hard on cubic graphs (Kong and Zhao in Congressus Numerantium 143:65–80, 2000), and the best approximation algorithm of maximum 2-independent set problem for cubic graphs has approximation ratio
8
15
(Miyano et al. in WALCOM 2017, Proceedings, pp 228–240). In this paper, we first prove that the maximum 3-star packing problem is NP-hard in claw-free cubic graphs and then design a linear-time algorithm which can find a 3-star packing of a connected claw-free cubic graph
G
covering at least
3
v
(
G
)
-
8
4
vertices, where
v
(
G
) denotes the number of vertices of
G
.</abstract><cop>New York</cop><pub>Springer US</pub><doi>10.1007/s10878-024-01115-z</doi><orcidid>https://orcid.org/0000-0002-4112-0469</orcidid></addata></record> |
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| subjects | Algorithms Apexes Approximation Combinatorics Convex and Discrete Geometry Graph theory Graphs Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Operations Research/Decision Theory Optimization Packing problem Theory of Computation |
| title | The maximum 3-star packing problem in claw-free cubic graphs |
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