Total Outer- Connected Dominating Sets and Total Outer- Connected Domination Polynomials of Cycle Graph ????????

Abstract

Let  be a simple graph of order . A dominating set is a total outer- connected dominating set of  if  is total dominating and the induced subgraph  is a connected graph. Let  be the family of total outer- connected dominating sets of a cycle graph   with cardinality  and let  In this paper we construct  and obtain exact formula and recursive formulae for . Using these formulae, we construct the polynomial , which we call the total outer- connected domination polynomial and obtain some properties of this polynomial.