Let G = (V,E) be a graph. A subset S ⊆ V is a dominating set of G if every vertex not in S is adjacent to a vertex in S. A set D̃ ⊆ V of a graph G = (V,E) is called an outer-connected dominating set for G if (1) D̃ is a dominating set for G, and (2) G[V \ D̃], the induced subgraph of G by V \ D̃, is connected. The minimum size among all outer-connected dominating sets of G is called the outerconnec...

A total outer-independent dominating set of a graph G = (V (G), E(G)) is a set D of vertices of G such that every vertex of G has a neighbor in D, and the set V (G) \D is independent. The total outer-independent domination number of a graph G, denoted by γ t (G), is the minimum cardinality of a total outer-independent dominating set of G. We prove that for every tree T of order n ≥ 4, with l le...

We obtain new results on the 2-rainbow domination number of generalized Petersen graphs P(ck,k). Exact values are established for all infinite families where general lower bound 45ck is attained. In other cases and upper bounds with small gaps given.