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

Pell graphs are defined on certain ternary strings as special subgraphs of Fibonacci cubes odd index. In this work the domination number, total 2-packing connected paired and signed number studied. Using integer linear programming, exact values some estimates for these numbers small obtained. Furthermore, theoretical bounds obtained graphs.

Let γt(G) and γpr(G) denote the total domination and the paired domination numbers of graph G, respectively, and let G ¤ H denote the Cartesian product of graphs G and H. In this paper, we show that γt(G)γt(H) ≤ 5γt(G ¤ H), which improves the known result γt(G)γt(H) ≤ 6γt(G ¤ H) given by Henning and Rall.

If G=(VG,EG) is a graph of order n, we call S⊆VG an isolating set if the induced by VG−NG[S] contains no edges. The minimum cardinality G called isolation number G, and it denoted ι(G). It known that ι(G)≤n3 bound sharp. A subset dominating in NG[S]=VG. domination number, γ(G). In this paper, analyze family trees T where ι(T)=γ(T), prove ι(T)=n3 implies ι(T)=γ(T). Moreover, give different equiv...

In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks 32 (1998) 199–206). A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by γpr(G), is the minimum cardinality of a p...