On Maximal Det-Independent (Res-Independent) Sets in Graphs
In this writing, we point out some errors made in Boutin (Graphs Combin 25:789–806, 2009), where the author claims that a maximal independent set in a hereditary system is a minimal determining (resolving) set. Further more, the author claims that if the exchange property holds at the level of minim...
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| Published in | Graphs and combinatorics Vol. 38; no. 2 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Tokyo
Springer Japan
01.04.2022
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0911-0119 1435-5914 1435-5914 |
| DOI | 10.1007/s00373-021-02452-0 |
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| Abstract | In this writing, we point out some errors made in Boutin (Graphs Combin 25:789–806, 2009), where the author claims that a maximal independent set in a hereditary system is a minimal determining (resolving) set. Further more, the author claims that if the exchange property holds at the level of minimal resolving sets, then, the corresponding hereditary system is a matroid. We give counter examples to disprove both of her claims. Besides, we prove that there exist graphs having such maximal independent sets which are not necessarily determining (resolving) sets. Also, we give necessary and sufficient conditions for a class of graphs to have a maximal independent set which is not minimal determining (resolving). |
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| AbstractList | In this writing, we point out some errors made in Boutin (Graphs Combin 25:789–806, 2009), where the author claims that a maximal independent set in a hereditary system is a minimal determining (resolving) set. Further more, the author claims that if the exchange property holds at the level of minimal resolving sets, then, the corresponding hereditary system is a matroid. We give counter examples to disprove both of her claims. Besides, we prove that there exist graphs having such maximal independent sets which are not necessarily determining (resolving) sets. Also, we give necessary and sufficient conditions for a class of graphs to have a maximal independent set which is not minimal determining (resolving). |
| ArticleNumber | 44 |
| Author | Salman, Muhammad Ali, Usman Zill-E-Shams |
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| Keywords | Resolving set Independent set Hereditary system 05C12 05B35 Determining set |
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| References | Ćaceres, Hernando, Mora, Pelayo, Puertas, Seara, Wood (CR3) 2007; 21 West (CR7) 1996 Chartrand, Eroh, Johnson, Oellermann (CR4) 2000; 105 Javaid, Fazil, Ali, Salman (CR6) 2018; 14 Boutin (CR1) 2009; 25 Hernando, Mora, Pelayo, Seera, Wood (CR5) 2010; 2010 Boutin (CR2) 2006; 13 G Chartrand (2452_CR4) 2000; 105 C Hernando (2452_CR5) 2010; 2010 J Ćaceres (2452_CR3) 2007; 21 DL Boutin (2452_CR1) 2009; 25 I Javaid (2452_CR6) 2018; 14 DB West (2452_CR7) 1996 DL Boutin (2452_CR2) 2006; 13 |
| References_xml | – volume: 14 start-page: 1 year: 2018 end-page: 12 ident: CR6 article-title: On some parameters related to fixing sets in graphs publication-title: J. Prime Res. Math. – volume: 2010 start-page: 17 year: 2010 ident: CR5 article-title: Extremal graph theory for metric dimension and diameter publication-title: Electron. J. Combin. – volume: 25 start-page: 789 year: 2009 end-page: 806 ident: CR1 article-title: Determining sets, resolving sets and the exchange property publication-title: Graphs Combin. doi: 10.1007/s00373-010-0880-6 – volume: 105 start-page: 99 issue: 1–3 year: 2000 end-page: 113 ident: CR4 article-title: Resolvability in graphs and the metric dimension of a graph publication-title: Discrete Appl. Math. doi: 10.1016/S0166-218X(00)00198-0 – volume: 13 start-page: 1 year: 2006 end-page: 12 ident: CR2 article-title: Identifying graph automorphism using determining sets publication-title: Electron. J. Combin. doi: 10.37236/1104 – volume: 21 start-page: 423 year: 2007 end-page: 441 ident: CR3 article-title: On the metric dimension of Cartesian products of graphs publication-title: SIAM J. Discrete Math. doi: 10.1137/050641867 – year: 1996 ident: CR7 publication-title: Introduction to Graph Theory – volume: 21 start-page: 423 year: 2007 ident: 2452_CR3 publication-title: SIAM J. Discrete Math. doi: 10.1137/050641867 – volume: 2010 start-page: 17 year: 2010 ident: 2452_CR5 publication-title: Electron. J. Combin. – volume-title: Introduction to Graph Theory year: 1996 ident: 2452_CR7 – volume: 14 start-page: 1 year: 2018 ident: 2452_CR6 publication-title: J. Prime Res. Math. – volume: 25 start-page: 789 year: 2009 ident: 2452_CR1 publication-title: Graphs Combin. doi: 10.1007/s00373-010-0880-6 – volume: 105 start-page: 99 issue: 1–3 year: 2000 ident: 2452_CR4 publication-title: Discrete Appl. Math. doi: 10.1016/S0166-218X(00)00198-0 – volume: 13 start-page: 1 year: 2006 ident: 2452_CR2 publication-title: Electron. J. Combin. doi: 10.37236/1104 |
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| Title | On Maximal Det-Independent (Res-Independent) Sets in Graphs |
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