On the double bondage number of graphs products

A set $D$ of vertices of graph $G$ is called $double$ $dominating$ $set$ if for any vertex $v$, $|N[v]\cap D|\geq 2$. The minimum cardinality of $double$ $domination$ of $G$ is denoted by $\gamma_d(G)$. The minimum number of edges $E'$ such that $\gamma_d(G\setminus E)>\gamma_d(G)$ is called the double bondage number of $G$ and is denoted by $b_d(G)$. This paper determines that $b_d(G\vee H)$ and exact values of $b(P_n\times P_2)$, and generalized corona product of graphs.