Isolation number: Cartesian and lexicographic products and generalized Sierpi\'{n}ski graphs

The isolation number $\iota(G)$ of a graph $G$ is the minimum cardinality of a set $A\subset V(G)$ such that the subgraph induced by the vertices that are not in the union of the closed neighborhoods of vertices in $A$ has no edges. The invariant, known also under the name vertex-edge domination number of $G$, has attracted a lot of interest in recent years. In this paper, we study the behavior of the isolation number under several graph operations, namely the Cartesian and the lexicographic product and the fractalization leading to generalized Sierpi\'{n}ski graphs. We prove several upper and lower bounds on the isolation number of the Cartesian product of two graphs. We prove a lower bound for the isolation number of the prism $G\,\Box\, K_2$ over an arbitrary graph $G$, which in the case of bipartite graphs leads to the equality $\iota(G\,\Box\, K_2)=\gamma(G)$, where $\gamma(G)$ is the domination number of $G$. In particular, $\iota(Q_{n+1})=\gamma(Q_n)$ holds for all positive integers $n$, where $Q_n$ is the $n$-dimensional hypercube. For the lexicographic product $G\circ H$ we prove that its isolation number, under certain mild restrictions, equals the total domination number of the first factor $G$. We also prove sharp lower and upper bounds on the isolation numbers of the generalized Sierpi\'{n}ski graphs $S_G^t$, where $G$ is an arbitrary base graph. These bounds in the case of classical Sierpi\'{n}ski graphs, namely $S_{K_n}^t$, coincide and lead to the exact values $\iota(S_{K_n}^t)=(n-1)\cdot n^{t-2}$ for all dimensions $t\ge 2$.