Let   be a simple graph in which V is the vertex set and E is the edge set. A total dominating set T of V is the set of vertices of  such that all its members have a neighbor in the set T. The total domination polynomial is calculated by the formula such that  where  is the number of total dominating sets of  of cardinality j. In the present paper, we determine total the domination polynomial of some interesting graphs.

total dominating set, total domination polynomial, caterpillar graph.