EDGE-CONNECTIVITY AUGMENTATION OF SIMPLE GRAPHS.
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| Title: | EDGE-CONNECTIVITY AUGMENTATION OF SIMPLE GRAPHS. |
|---|---|
| Authors: | JOHANSEN, KASPER SKOV1 kjoh@dtu.dk, ROTENBERG, EVA1 erot@dtu.dk, THOMASSEN, CARSTEN1 ctho@dtu.dk |
| Source: | SIAM Journal on Discrete Mathematics. 2025, Vol. 39 Issue 1, p163-169. 7p. |
| Subjects: | Polynomial time algorithms, Algorithms, Regular graphs |
| Abstract: | We consider the following variant of the edge-augmentation problem: Given a k-edge-connected graph with no loops or multiple edges, find a smallest edge set in the complement whose addition to G results in a (k + 1)-edge-connected graph. We establish the following dichotomy for this problem: If the complement of G contains a matching covering all vertices of G-degree k (and possibly more), then the complement also contains a matching whose addition to G results in a (k+1)-edge-connected graph. A smallest matching which augments the minimum degree can be found, in polynomial time, by Edmonds' matching algorithm, but it need not augment the edge-connectivity. Indeed, it is NP-hard to find a smallest edge-connectivity augmenting edge set, by a result of Tibor Jordán. On the other hand, if the complement of G contains no matching covering all vertices of G-degree k, then the complement has a minimum degree augmenting path system consisting of paths of length 1 or 2. Again we can find such a path system with as few edges as possible by Edmonds' matching algorithm. We can, in polynomial time, modify it to an edge-connectivity augmenting path system of paths of length 1 or 2 with the same number of edges, and this time it yields a smallest edge-connectivity augmenting set of edges. Combining these results, we conclude that a smallest edge-connectivity augmenting edge set in the complement of a k-regular, k-edge-connected simple graph has size n -- m(G), where n is the number of vertices of G, and m(G) is the size of a maximum matching in the complement of G. Another corollary is that the complement of every simple noncomplete graph G with n vertices has a set of at most 2n/3 edges whose addition to G results in a graph of larger edge-connectivity, with equality holding if and only the complement of G is a disjoint union of 3-cycles. [ABSTRACT FROM AUTHOR] |
| Copyright of SIAM Journal on Discrete Mathematics is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Items | – Name: Title Label: Title Group: Ti Data: EDGE-CONNECTIVITY AUGMENTATION OF SIMPLE GRAPHS. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22JOHANSEN%2C+KASPER+SKOV%22">JOHANSEN, KASPER SKOV</searchLink><relatesTo>1</relatesTo><i> kjoh@dtu.dk</i><br /><searchLink fieldCode="AR" term="%22ROTENBERG%2C+EVA%22">ROTENBERG, EVA</searchLink><relatesTo>1</relatesTo><i> erot@dtu.dk</i><br /><searchLink fieldCode="AR" term="%22THOMASSEN%2C+CARSTEN%22">THOMASSEN, CARSTEN</searchLink><relatesTo>1</relatesTo><i> ctho@dtu.dk</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22SIAM+Journal+on+Discrete+Mathematics%22">SIAM Journal on Discrete Mathematics</searchLink>. 2025, Vol. 39 Issue 1, p163-169. 7p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Polynomial+time+algorithms%22">Polynomial time algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Algorithms%22">Algorithms</searchLink><br /><searchLink fieldCode="DE" term="%22Regular+graphs%22">Regular graphs</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: We consider the following variant of the edge-augmentation problem: Given a k-edge-connected graph with no loops or multiple edges, find a smallest edge set in the complement whose addition to G results in a (k + 1)-edge-connected graph. We establish the following dichotomy for this problem: If the complement of G contains a matching covering all vertices of G-degree k (and possibly more), then the complement also contains a matching whose addition to G results in a (k+1)-edge-connected graph. A smallest matching which augments the minimum degree can be found, in polynomial time, by Edmonds' matching algorithm, but it need not augment the edge-connectivity. Indeed, it is NP-hard to find a smallest edge-connectivity augmenting edge set, by a result of Tibor Jordán. On the other hand, if the complement of G contains no matching covering all vertices of G-degree k, then the complement has a minimum degree augmenting path system consisting of paths of length 1 or 2. Again we can find such a path system with as few edges as possible by Edmonds' matching algorithm. We can, in polynomial time, modify it to an edge-connectivity augmenting path system of paths of length 1 or 2 with the same number of edges, and this time it yields a smallest edge-connectivity augmenting set of edges. Combining these results, we conclude that a smallest edge-connectivity augmenting edge set in the complement of a k-regular, k-edge-connected simple graph has size n -- m(G), where n is the number of vertices of G, and m(G) is the size of a maximum matching in the complement of G. Another corollary is that the complement of every simple noncomplete graph G with n vertices has a set of at most 2n/3 edges whose addition to G results in a graph of larger edge-connectivity, with equality holding if and only the complement of G is a disjoint union of 3-cycles. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of SIAM Journal on Discrete Mathematics is the property of Society for Industrial & Applied Mathematics and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1137/23M1574245 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 7 StartPage: 163 Subjects: – SubjectFull: Polynomial time algorithms Type: general – SubjectFull: Algorithms Type: general – SubjectFull: Regular graphs Type: general Titles: – TitleFull: EDGE-CONNECTIVITY AUGMENTATION OF SIMPLE GRAPHS. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: JOHANSEN, KASPER SKOV – PersonEntity: Name: NameFull: ROTENBERG, EVA – PersonEntity: Name: NameFull: THOMASSEN, CARSTEN IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 01 Text: 2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 08954801 Numbering: – Type: volume Value: 39 – Type: issue Value: 1 Titles: – TitleFull: SIAM Journal on Discrete Mathematics Type: main |
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