It is known that the removal of an edge from a graph G cannot decrease a domination number γ(G) and can increase it by at most one. Thus we can write that γ(G) ≤ γ(G-e) ≤ γ(G)+1 when an arbitrary edge e is removed. Here we present similar inequalities for the weakly connected domination number ${\gamma }_w$ and the connected domination number ${\gamma }_c$, i.e., we show that ${\gamma }_w\left(G\right)\le {\gamma }_w(G-e)\le {\gamma }_w\left(G\right)+1$ and ${\gamma }_c\left(G\right)\le {\gamma }_c(G-e)\le {\gamma }_c\left(G\right)+2$ if G and G-e are connected. Additionally we show that ${\gamma }_w\left(G\right)\le {\gamma }_w(G-Eₚ)\le {\gamma }_w\left(G\right)+p-1$ and ${\gamma }_c\left(G\right)\le {\gamma }_c(G-Eₚ)\le {\gamma }_c\left(G\right)+2p-2$ if G and G - Eₚ are connected and Eₚ = E(Hₚ) where Hₚ of order p is a connected...