Lower bounds on several versions of signed domination number

Three numerical invariants of graphs concerning domination, which are named the signed domination number γsγs, the k  -subdomination number γksγks and the signed total domination number γstγst, are studied in this paper. For any graph, some lower bounds on γsγs, γksγks and γstγst are presented, some of which generalize several known lower bounds on γsγs, γksγks and γstγst, while others are considered as new. It is also shown that these bounds are sharp.