Restricted domination number

Restricted domination number --- число ограниченного доминирования.

Let [math]\displaystyle{ U }[/math] be a subset of vertices of a graph [math]\displaystyle{ G }[/math]. The restricted domination number [math]\displaystyle{ r(G,U, \gamma) }[/math] of [math]\displaystyle{ U }[/math] is the minimum cardinality of a dominating set of [math]\displaystyle{ G }[/math] containing [math]\displaystyle{ U }[/math]. A smallest possible dominating set of [math]\displaystyle{ G }[/math] containing all the vertices in [math]\displaystyle{ U }[/math] is called a [math]\displaystyle{ \gamma_U }[/math]-set. The [math]\displaystyle{ k }[/math]-restricted domination number of [math]\displaystyle{ G }[/math] is the smallest integer [math]\displaystyle{ r_{k}(G,\gamma) }[/math] such that [math]\displaystyle{ r_{k}(G,U,\gamma) \leq r_{k}G,\gamma }[/math] for all subsets [math]\displaystyle{ U }[/math] of [math]\displaystyle{ V(G) }[/math] of cardinality [math]\displaystyle{ k }[/math]. In the case [math]\displaystyle{ k = 0 }[/math], the [math]\displaystyle{ k }[/math]-restricted dominaiton number is the domination number. When [math]\displaystyle{ k = 1 }[/math], the [math]\displaystyle{ k }[/math]-restricted domination number is called the domsaturation number of a graph and is denoted by [math]\displaystyle{ ds(G) }[/math].