Analysis on the characteristics for upper bound of [1,2]-domination in trees

In this paper, we propose a theoretical model for characterization and upper bounds of [1,2]-domination set of network which has tree structure. In detail, we propose a theoretic model for upper bounds on [1,2]-domination set of a tree network which has some typical constrains. To that purpose, we introduce a graph theory to model and analyze the characteristics of tree structure networks. We assume a node subset D of a graph G=(V,E). We define that D is a [1,2]-dominant set if for any node v in set V which is not an element of a set D is adjacent to a node or two nodes of an element in a set D (that is, 1 ≤ ∩ ≤ 2 for every node v ∈  ). The minimum cardinality of a [1,2]-dominating set of  , which is denoted by    , is called the [1,2]-domination number of  . In this paper, we show new upper bounds and characteristics about the [1,2]-domination number of tree.