Every tree and every twin-free unicyclic graph admits an independent locating-dominating set, while certain girth-4 graphs of order at least 9 do not.
Character- izations of trees with unique minimum locating-dominating sets
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Independent Locating-Dominating Sets in Pseudotrees
Every tree and every twin-free unicyclic graph admits an independent locating-dominating set, while certain girth-4 graphs of order at least 9 do not.