Let G be a graph with domination number γ(G). A dominating set S ⊆ V (G) has property UK if all components of the subgraph it induces in G are complete. The union of complete graphs domination number of a graph G, denoted γuk(G), is the minimum possible size of a dominating set of G, which has property UK. Results on changing and unchanging of γuk after vertex removal are presented. Also forbid...

A subset S ⊆ V (G) is an independent set of G if no two vertices of S are adjacent in G. A subset Q ⊆ V (G) is a 2-dominating set of G if each vertex from V (G)\Q has at least two neighbours in Q. We define new kind of kernels in graphs. Using existing concepts of an independent set and a 2-dominating set, we define in the natural way the concept of 2-dominating kernels in graphs. A subset J ⊂ ...