Differential in Cartesian Product Graphs

Let Gamma(V, E) be a graph of order n, S subset of V and let B(S) be the set of vertices in V \ S that have a neighbor in a set S. The differential of a set S is defined as partial derivative(S) = vertical bar B(S)vertical bar - vertical bar S vertical bar and the differential of the graph Gamma is defined as partial derivative(Gamma) = max{partial derivative(S) : S subset of V}. In this paper we obtain several tight bounds for the differential in Cartesian product graphs. In particular, we relate the differential in Cartesian product graphs with some known parameters of Gamma(1) x Gamma(2), namely, its domination number, its maximum and minimum degree and its order. Furthermore, we compute explicitly the differential of some class of product graphs.