Fault-diameter of Cartesian product of graphs and Cartesian graph bundles

Cartesian graph bundles is a class of graphs that is a generalization of the Cartesian graph products. Let ▫$G$▫ be a ▫$k_G$▫-connected graph and ▫${mathcal{D}}_c(G)$▫ denote the diameter of ▫$G$▫ after deleting any of its ▫$c < k_G$▫ vertices. We prove that if ▫$G_1, G_2, dots, G_q$▫ are ▫$k_1$▫-connected, ▫$k_2$▫-connected,...,▫$k_q$▫-connected graphs and ▫$0 leq a_1 < k_1$▫, ▫$0 leq a_2 < k_2$▫,...,▫$0 leq a_q < k_q$▫ and ▫$a = a_1 + a_2 + dots + a_q + (q-1)$▫, then the fault diameter of ▫$G$▫, a Cartesian product of ▫$G_1$▫, ▫$G_2$▫,...,▫$G_q$▫, with ▫$a$▫ faulty nodes is ▫${mathcal{D}}_{a}(G) leq {mathcal{D}}_{a_1}(G_1)+{mathcal{D}}_{a_2}(G_2) + dots + {mathcal{D}}_{a_q}(G_q) + 1$▫. We also show that ▫${mathcal{D}}_{a+b+1}(G) leq {mathcal{D}}_a(F) + {mathcal{D}}_b(B) + 1$▫ if ▫$G$▫ is a graph bundle with fibre ▫$F$▫ over base ▫$B$▫, ▫$a leq k_F$▫, and ▫$b leq k_B$▫. As an auxiliary result we prove that connectivity of graph bundle ▫$G$▫ is at least ▫$k_F+k_B$▫