We consider two general frameworks for multiple domination, which are called 〈r, s〉-domination and parametric domination. They generalise and unify {k}-domination, k-domination, total k-domination and k-tuple domination. In this paper, known upper bounds for the classical domination are generalised for the 〈r, s〉-domination and parametric domination numbers. These generalisations are based on t...

The Fibonacci cube Γn is the subgraph of the n-dimensional cube Qn induced by the vertices that contain no two consecutive 1s. Using integer linear programming, exact values are obtained for γt(Γn), n ≤ 12. Consequently, γt(Γn) ≤ 2Fn−10 + 21Fn−8 holds for n ≥ 11, where Fn are the Fibonacci numbers. It is proved that if n ≥ 9, then γt(Γn) ≥ d(Fn+2 − 11)/(n− 3)e−1. Using integer linear programmin...

For a positive integer k, a total {k}-dominating function of a graph G without isolated vertices is a function f from the vertex set V (G) to the set {0, 1, 2, . . . , k} such that for any vertex v ∈ V (G), the condition ∑ u∈N(v) f(u) ≥ k is fulfilled, where N(v) is the open neighborhood of v. The weight of a total {k}-dominating function f is the value ω(f) = ∑ v∈V f(v). The total {k}-dominati...

‎a set $s$ of vertices of a graph $g=(v,e)$ without isolated vertex‎ ‎is a {em total dominating set} if every vertex of $v(g)$ is‎ ‎adjacent to some vertex in $s$‎. ‎the {em total domatic number} of‎ ‎a graph $g$ is the maximum number of total dominating sets into‎ ‎which the vertex set of $g$ can be partitioned‎. ‎we show that the‎ ‎total domatic number of a random $r$-regular graph is almost‎...

‎In this paper‎, ‎we investigate domination number‎, ‎$gamma$‎, ‎as well‎ ‎as signed domination number‎, ‎$gamma_{_S}$‎, ‎of all cubic Cayley‎ ‎graphs of cyclic and quaternion groups‎. ‎In addition‎, ‎we show that‎ ‎the domination and signed domination numbers of cubic graphs depend‎ on each other‎.