The g-extra connectivity of graph products

Connectivity is one of important parameters for the fault tolerant of an interconnection network. In 1996, Fàbrega and Fiol proposed the concept of g-extra connectivity. A subset of vertices S is said to be a cutset if $G-S$ is not connected. A cutset S is called an $R_g$-cutset, where g is a non-negative integer, if every component of $G-S$ has at least $g+1$ vertices. If G has at least one $R_g$-cutset, the g-extra connectivity of G, denoted by ${\kappa }_g\left(G\right)$, is then defined as the minimum cardinality over all $R_g$-cutsets of G. In this paper, we first obtain the exact value of g-extra connectivity for the lexicographic product of two general graphs. Next, the upper and lower sharp bounds of g-extra connectivity for the Cartesian product of two general graphs are given. In the end, we apply our results on grid graphs and 2-dimensional generalized hypercubes.