total outer-connected dominating sets and total outer-connected domination polynomial of complete bipartite graph k2,n

Let g = (v, e) be a simple graph. a set d⊆v (g) is a total outer−connected dominating set of g if d is total dominating, and the induced sub-graph g[v (g) − d] is a connected graph. let k2,n be the complete bipartite graph and d ̃tc (k2,n,i) denote the family of all total outer-connected dominating sets of k2,n with cardinality i. let d ̃tc (k2,n,i)=|d ̃tc (k2,n,i)|. in this paper, we obtain recursive formula for d ̃tc (k2,n,i). using this recursive formula, we construct the polynomial, d~tc (k2,n,x)=∑i=22+nd ̃tc (k2,n,i)xi  which we call total outer−connected domination polynomial of k2,n and obtain some  properties of this polynomial.