Dominating Sets with Neighborhoods for Trees

Subset V 1pt ' V (G) is called a -dominating set of vertices of graph G with neighborhood if for any vertex v V V 1pt ' there is vertex u V 1pt ' such that the length of the shortest path connecting these vertices ~d (u, v{\;}) ~\, \, \, \, ; { } (G) is the number of vertices in the minimal -dominating set; { } (G) = 1 for ~r (G) ~~~ d (G) ; for 1, but the calculation of { ₁} (G) = (G) is an NP‑complete problem. This paper considers class of trees t₃^ of diameter d whose degrees of all internal vertices are equal to. Constructive descriptions of trees t t₃^ are given. Procedures are developed for computing the values of { } (t) in the range ~~1 < r (t). Asymptotic estimates are established for { } (t) and their proportion of the total number of vertices in t t₃^ as d. Computational examples are given.