On the super domination number of lexicographic product graphs

The neighbourhood of a vertex upsilon of a graph G is the set N(upsilon) of all vertices adjacent to upsilon in G. For D subset of V(G) we define (D) over bar = V(G) \ D. A set D subset of V(G) is called a super dominating set if for every vertex u is an element of (D) over bar, there exists upsilon is an element of D such that N(upsilon) boolean AND (D) over bar = {u}. The super domination number of G is the minimum cardinality among all super dominating sets in G. In this article we obtain closed formulas and tight bounds for the super dominating number of lexicographic product graphs in terms of invariants of the factor graphs involved in the product. As a consequence of the study, we show that the problem of finding the super domination number of a graph is NP-Hard. (C) 2018 Elsevier B.V. All rights reserved.