Let  be a simple graph. A set  is a dominating (total dominating) set of G if for every   there exists  such that  A dominating (total dominating) set S of G is a secure dominating (resp. secure total dominating) set of G if for each  there exists  such that  is a dominating (resp. total dominating) set of G. In this paper, we characterize the secure dominating and the secure total dominating sets in the lexicographic product of connected graphs. As a consequence, we determine some (sharp) bounds for the secure domination and secure total domination numbers of the lexicographic product.

domination, total domination, secure domination, secure total domination, lexicographic product.