Bounds on the signed distance-􀝇-domination number of graphs

Let ?? ?? ????, ???? be a graph with vertex set ?? ?? ???????? of order ?? and edge set ?? ?? ????????. A ?? -dominating set of ?? is a subset ?? ?? ?? such that each vertex in ?? \ ?? has at least ?? neighbors in ??. If ?? is a vertex of a graph ??, the open ??-neighborhood of ??, denoted by ??????????, is the set ?????????? ?? ???? ?? ?? ?? ?? ?? ?? ?????? ??????, ???? ?? ?? ??. ?????????? ?? ?????????? ?? ?????? is the closed ??-neighborhood of ??. A function ?? ?? ?? ?? ????1, 1?? is a signed distance-??-dominating function of ??, if for every vertex ?? ?? ??, ???????????????? ?? ??? ?? ?????????? ???????? ?? 1. The signed distance-??-domination number, denoted by ????,????????, is the minimum weight of a signed distance-??-dominating function of ??. In this paper, we give lower and upper bounds on ????,?? of graphs. Also, we determine the signed distance-??-domination number of graph ????,?????? ?? ???? (the graph obtained from the disjoint union ?? ?? ?? by adding the edges ?????? ?? ?? ?? ????????, ?? ?? ??????????) when ?? ?? 2.