An extension of locating-total domination problem and its complexity

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An r-dominating set (r-total dominating set) of G is a subset S of V (G) for which Nr(u)∩S is non-empty for all u not in S (for all u in V (G)). An r-locating-dominating set (r-locating-total dominating set) of G is an r-dominating set (r-total dominating set) S of G for which Nr(u) ∩ S is different from Nr(v) ∩ S for all u and v not in S. This paper presents an extension of the locating-total dominating set of G. Further, we establish a lower bound on r-locating-dominating set and r-locating-total dominating set for k-regular graphs, as well as demonstrate that r-locating-total dominating set is an NP-complete problem. Furthermore, the r-locating-dominating set and r-locating-total dominating set problems are discussed for some standard graphs.