Abstract Given a graph G and subset of vertices $$D\subseteq V(G)$$ D ⊆ V ( G ) , the external neighbourhood D is defined as $$N_e(D)=\{u\in V(G){\setminus } D:\, N(u)\cap D\ne \varnothing \}$$ <mml:msu...

We consider four different types of multiple domination and provide new improved upper bounds for the kand k-tuple domination numbers. They generalise two classical bounds for the domination number and are better than a number of known upper bounds for these two multiple domination parameters. Also, we explicitly present and systematize randomized algorithms for finding multiple dominating sets...