Graphs with the Strong Havel–Hakimi Property
The Havel–Hakimi algorithm iteratively reduces the degree sequence of a graph to a list of zeroes. As shown by Favaron, Mahéo, and Saclé, the number of zeroes produced, known as the residue, is a lower bound on the independence number of the graph. We say that a graph has the strong Havel–Hakimi pro...
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| Published in | Graphs and combinatorics Vol. 32; no. 5; pp. 1689 - 1697 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Tokyo
Springer Japan
01.09.2016
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0911-0119 1435-5914 |
| DOI | 10.1007/s00373-015-1674-7 |
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| Abstract | The Havel–Hakimi algorithm iteratively reduces the degree sequence of a graph to a list of zeroes. As shown by Favaron, Mahéo, and Saclé, the number of zeroes produced, known as the residue, is a lower bound on the independence number of the graph. We say that a graph has the strong Havel–Hakimi property if in each of its induced subgraphs, deleting any vertex of maximum degree reduces the degree sequence in the same way that the Havel–Hakimi algorithm does. We characterize graphs having this property (which include all threshold and matrogenic graphs) in terms of minimal forbidden induced subgraphs. We further show that for these graphs the residue equals the independence number, and a natural greedy algorithm always produces a maximum independent set. |
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| AbstractList | The Havel–Hakimi algorithm iteratively reduces the degree sequence of a graph to a list of zeroes. As shown by Favaron, Mahéo, and Saclé, the number of zeroes produced, known as the residue, is a lower bound on the independence number of the graph. We say that a graph has the strong Havel–Hakimi property if in each of its induced subgraphs, deleting any vertex of maximum degree reduces the degree sequence in the same way that the Havel–Hakimi algorithm does. We characterize graphs having this property (which include all threshold and matrogenic graphs) in terms of minimal forbidden induced subgraphs. We further show that for these graphs the residue equals the independence number, and a natural greedy algorithm always produces a maximum independent set. |
| Author | Barrus, Michael D. Molnar, Grant |
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| Cites_doi | 10.1016/j.disc.2013.12.013 10.1137/0110037 10.1002/jgt.3190150107 10.1002/(SICI)1097-0118(199605)22:1<89::AID-JGT12>3.0.CO;2-J 10.1016/0012-365X(92)00479-B |
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| Keywords | 05C07 05C69 Havel–Hakimi algorithm Residue Independence number |
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| References | Triesch (CR11) 1996; 22 Griggs, Kleitman (CR6) 1994; 127 CR5 CR8 Amos, Davila, Pepper (CR1) 2014; 321 Favaron, Mahéo, Saclé (CR4) 1991; 15 Fajtlowicz (CR3) 1988; 66 Mahadev, Peled (CR10) 1995 Jelen (CR9) 1999; 127 Chartrand, Lesniak, Zhang (CR2) 2011 Hakimi (CR7) 1962; 10 S Fajtlowicz (1674_CR3) 1988; 66 1674_CR8 O Favaron (1674_CR4) 1991; 15 NVR Mahadev (1674_CR10) 1995 1674_CR5 J Griggs (1674_CR6) 1994; 127 G Chartrand (1674_CR2) 2011 S Hakimi (1674_CR7) 1962; 10 E Triesch (1674_CR11) 1996; 22 F Jelen (1674_CR9) 1999; 127 D Amos (1674_CR1) 2014; 321 |
| References_xml | – year: 2011 ident: CR2 publication-title: Graphs & Digraphs – volume: 321 start-page: 24 year: 2014 end-page: 34 ident: CR1 article-title: On the -residue of disjoint unions of graphs with applications to -independence publication-title: Discrete Math. doi: 10.1016/j.disc.2013.12.013 – volume: 10 start-page: 496 year: 1962 end-page: 506 ident: CR7 article-title: On the realizability of a set of integers as degree sequences of the vertices of a graph publication-title: J. SIAM Appl. Math. doi: 10.1137/0110037 – ident: CR5 – volume: 15 start-page: 39 issue: 1 year: 1991 end-page: 64 ident: CR4 article-title: On the residue of a graph publication-title: J. Graph Theory doi: 10.1002/jgt.3190150107 – ident: CR8 – volume: 127 start-page: 209 year: 1999 end-page: 212 ident: CR9 article-title: -Independence and the -residue of a graph publication-title: J. Graph Theory – volume: 66 start-page: 23 year: 1988 end-page: 32 ident: CR3 article-title: On conjectures of Graffiti, III publication-title: Cong. Numer. – year: 1995 ident: CR10 publication-title: Threshold graphs and related topics. Annals of Discrete Mathematics, 56 – volume: 22 start-page: 89 issue: 1 year: 1996 end-page: 93 ident: CR11 article-title: Degree sequences of graphs and dominance order publication-title: J. Graph Theory doi: 10.1002/(SICI)1097-0118(199605)22:1<89::AID-JGT12>3.0.CO;2-J – volume: 127 start-page: 209 issue: 1–3 year: 1994 end-page: 212 ident: CR6 article-title: Independence and the Havel–Hakimi residue, Graph theory and applications (Hakone, 1990) publication-title: Discrete Math. doi: 10.1016/0012-365X(92)00479-B – ident: 1674_CR8 – volume: 22 start-page: 89 issue: 1 year: 1996 ident: 1674_CR11 publication-title: J. Graph Theory doi: 10.1002/(SICI)1097-0118(199605)22:1<89::AID-JGT12>3.0.CO;2-J – ident: 1674_CR5 – volume: 66 start-page: 23 year: 1988 ident: 1674_CR3 publication-title: Cong. Numer. – volume: 321 start-page: 24 year: 2014 ident: 1674_CR1 publication-title: Discrete Math. doi: 10.1016/j.disc.2013.12.013 – volume: 127 start-page: 209 year: 1999 ident: 1674_CR9 publication-title: J. Graph Theory – volume-title: Threshold graphs and related topics. Annals of Discrete Mathematics, 56 year: 1995 ident: 1674_CR10 – volume: 10 start-page: 496 year: 1962 ident: 1674_CR7 publication-title: J. SIAM Appl. Math. doi: 10.1137/0110037 – volume: 15 start-page: 39 issue: 1 year: 1991 ident: 1674_CR4 publication-title: J. Graph Theory doi: 10.1002/jgt.3190150107 – volume-title: Graphs & Digraphs year: 2011 ident: 1674_CR2 – volume: 127 start-page: 209 issue: 1–3 year: 1994 ident: 1674_CR6 publication-title: Discrete Math. doi: 10.1016/0012-365X(92)00479-B |
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| Snippet | The Havel–Hakimi algorithm iteratively reduces the degree sequence of a graph to a list of zeroes. As shown by Favaron, Mahéo, and Saclé, the number of zeroes... |
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| Title | Graphs with the Strong Havel–Hakimi Property |
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