Assume we have a set of k colors and we assign an arbitrary subset of these colors to each vertex of a graph G. If we require that each vertex to which an empty set is assigned has in its neighborhood all k colors, then this assignment is called a k-rainbow dominating function of G. The corresponding invariant γrk(G), which is the minimum sum of numbers of assigned colors over all vertices of G...

a dominating set $d subseteq v$ of a graph $g = (v,e)$ is said to be a connected cototal dominating set if $langle d rangle$ is connected and $langle v-d rangle neq phi$, contains no isolated vertices. a connected cototal dominating set is said to be minimal if no proper subset of $d$ is connected cototal dominating set. the connected cototal domination number $gamma_{ccl}(g)$ of $g$ is the min...

We obtain new results on 3-rainbow domination numbers of generalized Petersen graphs P(6k,k). In some cases, for infinite families, exact values are established; in all other the lower and upper bounds with small gaps given. also define singleton rainbow domination, where sets assigned have a cardinality of, at most, one, provide analogous this special case domination.

A set S of vertices of a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. The independent domination number of G, denoted by i(G), is the minimum cardinality of an independent dominating set of G. In this paper, some new classes of graphs with equal domination and independent domination numbers are presented and exa...