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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Zill-E-Shams, Salman, M. & Ali, U. On Maximal Det-Independent (Res-Independent) Sets in Graphs. Graphs and Combinatorics 38, 44 (2022). https://doi.org/10.1007/s00373-021-02452-0
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DOI: https://doi.org/10.1007/s00373-021-02452-0