Nordhaus–Gaddum-Type Results on the Connected Edge Domination Number
A connected edge dominating set of a connected graph G = ( V , E ) is a subset X of E such that the edge-induced subgraph G [ X ] is connected and each e ∈ E ( G ) \ X has at least one neighbor in X . The connected edge domination number γ c ′ ( G ) of G is the minimum cardinality of a connected edg...
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| Veröffentlicht in: | Bulletin of the Malaysian Mathematical Sciences Society 2024-03, Vol.47 (2), Article 54 |
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| container_title | Bulletin of the Malaysian Mathematical Sciences Society |
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| creator | Li, Hengzhe Liu, Huayue Liu, Jianbing |
| description | A
connected edge dominating set
of a connected graph
G
=
(
V
,
E
)
is a subset
X
of
E
such that the edge-induced subgraph
G
[
X
] is connected and each
e
∈
E
(
G
)
\
X
has at least one neighbor in
X
. The
connected edge domination number
γ
c
′
(
G
)
of
G
is the minimum cardinality of a connected edge dominating set of
G
. An edge dominating set
X
of a graph
G
is called a 2
-edge-connected edge dominating set
if
G
[
X
] is 2-edge-connected. The 2
-edge-connected edge dominating number
γ
2
e
c
′
(
G
)
is the minimum size of a 2-edge-connected edge dominating set of
G
. In this paper, we obtain the sharp lower bounds for
γ
c
′
(
G
)
+
γ
c
′
(
G
¯
)
and
γ
c
′
(
G
)
·
γ
c
′
(
G
¯
)
. Moreover, we characterize the classes of graphs attaining the lower bounds and study the relationship between
γ
c
′
(
G
)
and several other parameters, such as independent number and vertex cover number. In addition, we show that
3
≤
γ
2
e
c
′
(
G
)
≤
⌊
3
2
(
n
-
1
)
⌋
if
G
is 2-edge-connected. We also obtain the upper and lower bounds for
γ
2
e
c
′
(
G
)
+
γ
2
e
c
′
(
G
¯
)
and
γ
2
e
c
′
(
G
)
·
γ
2
e
c
′
(
G
¯
)
and characterize the classes of graphs attaining the lower bounds. |
| doi_str_mv | 10.1007/s40840-023-01649-1 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_2922540929</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>2922540929</sourcerecordid><originalsourceid>FETCH-LOGICAL-c270t-772fc48ee1d21cd8028e0d984c5aad6436301a422aa7c871ec6cb05dfef7056c3</originalsourceid><addsrcrecordid>eNp9kE1OwzAQhS0EElXpBVhFYm0YTxwnWaLSFqSqSKisLdee9EdtUuxk0R134IacBEOQ2DGbWcx7b54-xq4F3AqA_C5IKCRwwJSDULLk4owNUBTAJYI6ZwMQqLjKIbtkoxB2ECdTqFAM2HTReLcxXfh8_5gZ57oDX56OlLxQ6PZtSJo6aTeUjJu6JtuSSyZuTclDc9jWpt3G66I7rMhfsYvK7AONfveQvU4ny_Ejnz_Pnsb3c24xh5bnOVZWFkTCobCuACwIXFlImxnjlExVCsJIRGNyW-SCrLIryFxFVWyvbDpkN33u0TdvHYVW75rO1_GlxhIxk1BiGVXYq6xvQvBU6aPfHow_aQH6G5nukemITP8g0yKa0t4Uorhek_-L_sf1BYFuboU</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>2922540929</pqid></control><display><type>article</type><title>Nordhaus–Gaddum-Type Results on the Connected Edge Domination Number</title><source>SpringerLink Journals - AutoHoldings</source><creator>Li, Hengzhe ; Liu, Huayue ; Liu, Jianbing</creator><creatorcontrib>Li, Hengzhe ; Liu, Huayue ; Liu, Jianbing</creatorcontrib><description>A
connected edge dominating set
of a connected graph
G
=
(
V
,
E
)
is a subset
X
of
E
such that the edge-induced subgraph
G
[
X
] is connected and each
e
∈
E
(
G
)
\
X
has at least one neighbor in
X
. The
connected edge domination number
γ
c
′
(
G
)
of
G
is the minimum cardinality of a connected edge dominating set of
G
. An edge dominating set
X
of a graph
G
is called a 2
-edge-connected edge dominating set
if
G
[
X
] is 2-edge-connected. The 2
-edge-connected edge dominating number
γ
2
e
c
′
(
G
)
is the minimum size of a 2-edge-connected edge dominating set of
G
. In this paper, we obtain the sharp lower bounds for
γ
c
′
(
G
)
+
γ
c
′
(
G
¯
)
and
γ
c
′
(
G
)
·
γ
c
′
(
G
¯
)
. Moreover, we characterize the classes of graphs attaining the lower bounds and study the relationship between
γ
c
′
(
G
)
and several other parameters, such as independent number and vertex cover number. In addition, we show that
3
≤
γ
2
e
c
′
(
G
)
≤
⌊
3
2
(
n
-
1
)
⌋
if
G
is 2-edge-connected. We also obtain the upper and lower bounds for
γ
2
e
c
′
(
G
)
+
γ
2
e
c
′
(
G
¯
)
and
γ
2
e
c
′
(
G
)
·
γ
2
e
c
′
(
G
¯
)
and characterize the classes of graphs attaining the lower bounds.</description><identifier>ISSN: 0126-6705</identifier><identifier>EISSN: 2180-4206</identifier><identifier>DOI: 10.1007/s40840-023-01649-1</identifier><language>eng</language><publisher>Singapore: Springer Nature Singapore</publisher><subject>Applications of Mathematics ; Graph theory ; Graphs ; Lower bounds ; Mathematics ; Mathematics and Statistics</subject><ispartof>Bulletin of the Malaysian Mathematical Sciences Society, 2024-03, Vol.47 (2), Article 54</ispartof><rights>The Author(s), under exclusive licence to Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 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-772fc48ee1d21cd8028e0d984c5aad6436301a422aa7c871ec6cb05dfef7056c3</cites><orcidid>0000-0001-6004-971X</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/s40840-023-01649-1$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s40840-023-01649-1$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,776,780,27901,27902,41464,42533,51294</link.rule.ids></links><search><creatorcontrib>Li, Hengzhe</creatorcontrib><creatorcontrib>Liu, Huayue</creatorcontrib><creatorcontrib>Liu, Jianbing</creatorcontrib><title>Nordhaus–Gaddum-Type Results on the Connected Edge Domination Number</title><title>Bulletin of the Malaysian Mathematical Sciences Society</title><addtitle>Bull. Malays. Math. Sci. Soc</addtitle><description>A
connected edge dominating set
of a connected graph
G
=
(
V
,
E
)
is a subset
X
of
E
such that the edge-induced subgraph
G
[
X
] is connected and each
e
∈
E
(
G
)
\
X
has at least one neighbor in
X
. The
connected edge domination number
γ
c
′
(
G
)
of
G
is the minimum cardinality of a connected edge dominating set of
G
. An edge dominating set
X
of a graph
G
is called a 2
-edge-connected edge dominating set
if
G
[
X
] is 2-edge-connected. The 2
-edge-connected edge dominating number
γ
2
e
c
′
(
G
)
is the minimum size of a 2-edge-connected edge dominating set of
G
. In this paper, we obtain the sharp lower bounds for
γ
c
′
(
G
)
+
γ
c
′
(
G
¯
)
and
γ
c
′
(
G
)
·
γ
c
′
(
G
¯
)
. Moreover, we characterize the classes of graphs attaining the lower bounds and study the relationship between
γ
c
′
(
G
)
and several other parameters, such as independent number and vertex cover number. In addition, we show that
3
≤
γ
2
e
c
′
(
G
)
≤
⌊
3
2
(
n
-
1
)
⌋
if
G
is 2-edge-connected. We also obtain the upper and lower bounds for
γ
2
e
c
′
(
G
)
+
γ
2
e
c
′
(
G
¯
)
and
γ
2
e
c
′
(
G
)
·
γ
2
e
c
′
(
G
¯
)
and characterize the classes of graphs attaining the lower bounds.</description><subject>Applications of Mathematics</subject><subject>Graph theory</subject><subject>Graphs</subject><subject>Lower bounds</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><issn>0126-6705</issn><issn>2180-4206</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kE1OwzAQhS0EElXpBVhFYm0YTxwnWaLSFqSqSKisLdee9EdtUuxk0R134IacBEOQ2DGbWcx7b54-xq4F3AqA_C5IKCRwwJSDULLk4owNUBTAJYI6ZwMQqLjKIbtkoxB2ECdTqFAM2HTReLcxXfh8_5gZ57oDX56OlLxQ6PZtSJo6aTeUjJu6JtuSSyZuTclDc9jWpt3G66I7rMhfsYvK7AONfveQvU4ny_Ejnz_Pnsb3c24xh5bnOVZWFkTCobCuACwIXFlImxnjlExVCsJIRGNyW-SCrLIryFxFVWyvbDpkN33u0TdvHYVW75rO1_GlxhIxk1BiGVXYq6xvQvBU6aPfHow_aQH6G5nukemITP8g0yKa0t4Uorhek_-L_sf1BYFuboU</recordid><startdate>20240301</startdate><enddate>20240301</enddate><creator>Li, Hengzhe</creator><creator>Liu, Huayue</creator><creator>Liu, Jianbing</creator><general>Springer Nature Singapore</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0001-6004-971X</orcidid></search><sort><creationdate>20240301</creationdate><title>Nordhaus–Gaddum-Type Results on the Connected Edge Domination Number</title><author>Li, Hengzhe ; Liu, Huayue ; Liu, Jianbing</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-772fc48ee1d21cd8028e0d984c5aad6436301a422aa7c871ec6cb05dfef7056c3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Applications of Mathematics</topic><topic>Graph theory</topic><topic>Graphs</topic><topic>Lower bounds</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Li, Hengzhe</creatorcontrib><creatorcontrib>Liu, Huayue</creatorcontrib><creatorcontrib>Liu, Jianbing</creatorcontrib><collection>CrossRef</collection><jtitle>Bulletin of the Malaysian Mathematical Sciences Society</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Li, Hengzhe</au><au>Liu, Huayue</au><au>Liu, Jianbing</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Nordhaus–Gaddum-Type Results on the Connected Edge Domination Number</atitle><jtitle>Bulletin of the Malaysian Mathematical Sciences Society</jtitle><stitle>Bull. Malays. Math. Sci. Soc</stitle><date>2024-03-01</date><risdate>2024</risdate><volume>47</volume><issue>2</issue><artnum>54</artnum><issn>0126-6705</issn><eissn>2180-4206</eissn><abstract>A
connected edge dominating set
of a connected graph
G
=
(
V
,
E
)
is a subset
X
of
E
such that the edge-induced subgraph
G
[
X
] is connected and each
e
∈
E
(
G
)
\
X
has at least one neighbor in
X
. The
connected edge domination number
γ
c
′
(
G
)
of
G
is the minimum cardinality of a connected edge dominating set of
G
. An edge dominating set
X
of a graph
G
is called a 2
-edge-connected edge dominating set
if
G
[
X
] is 2-edge-connected. The 2
-edge-connected edge dominating number
γ
2
e
c
′
(
G
)
is the minimum size of a 2-edge-connected edge dominating set of
G
. In this paper, we obtain the sharp lower bounds for
γ
c
′
(
G
)
+
γ
c
′
(
G
¯
)
and
γ
c
′
(
G
)
·
γ
c
′
(
G
¯
)
. Moreover, we characterize the classes of graphs attaining the lower bounds and study the relationship between
γ
c
′
(
G
)
and several other parameters, such as independent number and vertex cover number. In addition, we show that
3
≤
γ
2
e
c
′
(
G
)
≤
⌊
3
2
(
n
-
1
)
⌋
if
G
is 2-edge-connected. We also obtain the upper and lower bounds for
γ
2
e
c
′
(
G
)
+
γ
2
e
c
′
(
G
¯
)
and
γ
2
e
c
′
(
G
)
·
γ
2
e
c
′
(
G
¯
)
and characterize the classes of graphs attaining the lower bounds.</abstract><cop>Singapore</cop><pub>Springer Nature Singapore</pub><doi>10.1007/s40840-023-01649-1</doi><orcidid>https://orcid.org/0000-0001-6004-971X</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0126-6705 |
| ispartof | Bulletin of the Malaysian Mathematical Sciences Society, 2024-03, Vol.47 (2), Article 54 |
| issn | 0126-6705 2180-4206 |
| language | eng |
| recordid | cdi_proquest_journals_2922540929 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Applications of Mathematics Graph theory Graphs Lower bounds Mathematics Mathematics and Statistics |
| title | Nordhaus–Gaddum-Type Results on the Connected Edge Domination Number |
| url | https://sfx.bib-bvb.de/sfx_tum?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-28T23%3A17%3A14IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-proquest_cross&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Nordhaus%E2%80%93Gaddum-Type%20Results%20on%20the%20Connected%20Edge%20Domination%20Number&rft.jtitle=Bulletin%20of%20the%20Malaysian%20Mathematical%20Sciences%20Society&rft.au=Li,%20Hengzhe&rft.date=2024-03-01&rft.volume=47&rft.issue=2&rft.artnum=54&rft.issn=0126-6705&rft.eissn=2180-4206&rft_id=info:doi/10.1007/s40840-023-01649-1&rft_dat=%3Cproquest_cross%3E2922540929%3C/proquest_cross%3E%3Curl%3E%3C/url%3E&disable_directlink=true&sfx.directlink=off&sfx.report_link=0&rft_id=info:oai/&rft_pqid=2922540929&rft_id=info:pmid/&rfr_iscdi=true |