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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Bibliographic Details
Published inGraphs and combinatorics Vol. 38; no. 2
Main Authors Zill-E-Shams, Salman, Muhammad, Ali, Usman
Format Journal Article
LanguageEnglish
Published Tokyo Springer Japan 01.04.2022
Springer Nature B.V
Subjects
Combinatorics
Engineering Design
Graphs
Mathematics
Mathematics and Statistics
Original Paper
Resolving set
Independent set
Hereditary system
05C12
05B35
Determining set
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ISSN0911-0119
1435-5914
1435-5914
DOI10.1007/s00373-021-02452-0

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Summary: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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ISSN:0911-0119
1435-5914
1435-5914
DOI:10.1007/s00373-021-02452-0