Domination and 2-domination numbers are defined only for graphs with non-isolated vertices. In a Graph G = (V, E) each vertex is said to dominate every in its closed neighborhood. graph G, subset S of V(G) called 2-dominating set if v ∈ V, V-S has atleast two neighbors S. The smallest cardinality known as the number γ2(G). this paper, we find some special also graphs.

A caterpillar is a tree with the property that after deleting all its vertices of degree 1 a simple path is obtained. The signed 2-domination number γ s (G) and the signed total 2-domination number γ st(G) of a graph G are variants of the signed domination number γs(G) and the signed total domination number γst(G). Their values for caterpillars are studied.