Lower bounds on the signed (total) $k$-domination number

Let G be a graph with vertex set V (G). For any integer k ≥ 1, a signed (total) k-dominating function is a function f : V (G) → {−1, 1} satisfying f (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.