Lower Bounds on the Signed (Total) $K$-Domination Number

Let g be a graph with vertex set v (g). for any integer k ≥ 1, asigned (total) k-dominating function is a function f : v (g) → {−1, 1} satisfyingf (x) k ( f (x) k) for every v v (g), where n (v) is the neigh- borhood of v and n [v] = n (v) ∪ {v}. the minimum of the values v∈v (g) f (v),taken over all signed (total) k-dominating functions f , is called the signed (total) k-domination number. the clique number of a graph g is the maximum cardinality of a complete subgraph of g. in this note we present some new sharp lower bounds on the signed (total) k-domination number depending on the clique number of the graph.lts improve some known bounds.