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 γw and the connected domination number γc, i.e., we show that γw(G) ≤ γw(G− e) ≤ γw(G) + 1 and γc(G) ≤ γc(G − e) ≤ γc(G) + 2 if G and G − e are connected. Additionally we show that γw(G) ≤ γw(G−Ep) ≤ γw(G) + p− 1 and γc(G) ≤ γc(G− Ep) ≤ γc(G) + 2p− 2 if G and G− Ep are connected and Ep = E(Hp) where Hp of order p is a connected subgraph of G.