The Compared Costs of Domination Location-Domination and Identification - Hudry, Olivier; Lobstein, Antoine (Discussiones Mathematicae. Graph Theory, 2020)
Let = ( ) be a finite graph and ≥ 1 be an integer. For ∈ , let ) = { ∈ : ) ≤ } be the ball of radius centered at . A set ⊆ is an -dominating code if for all ∈ , we have ) ∩ ≠ ∅; it is an -locating-dominating code if for all ∈ , we have ) ∩ ≠ ∅, and for any two distinct non-codewords ∈ \ , ∈ \ , we have ) ∩ ≠ ) ∩ ; it is an -identifying code if for all ∈ , we have ) ∩ ≠ ∅, and for any two distinct vertices ∈ , ∈ , we have ) ∩ ≠ ) ∩ . We denote by γ ) (respectively, ) and )) the smallest possible cardinality of an -dominating code (respectively, an -locating-dominating code and an -identifying code). We study how small and how large the three differences )− ), )−γ ) and ) − γ ) can be.