Domination numbers in graphs with removed edge or set of edges

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 ${\displaystyle {\gamma }_w}$ and the connected domination number ${\displaystyle {\gamma }_c}$, i.e., we show that ${\displaystyle {\gamma }_w\left(G\right)\le {\gamma }_w(G-e)\le {\gamma }_w\left(G\right)+1}$ and ${\displaystyle {\gamma }_c\left(G\right)\le {\gamma }_c(G-e)\le {\gamma }_c\left(G\right)+2}$ if G and G-e are connected. Additionally we show that ${\displaystyle {\gamma }_w\left(G\right)\le {\gamma }_w(G-Eₚ)\le {\gamma }_w\left(G\right)+p-1}$ and ${\displaystyle {\gamma }_c\left(G\right)\le {\gamma }_c(G-Eₚ)\le {\gamma }_c\left(G\right)+2p-2}$ if G and G - Eₚ are connected and Eₚ = E(Hₚ) where Hₚ of order p is a connected subgraph of G.