Twain Secure Perfect Dominating Sets and Twain Secure Perfect Domination Polynomials of Stars

Abstract

Let  be a simple graph. A set  is a dominating set of  if for every vertex in  is adjacent to atleast one vertex in  A subset  of  is called a twain secure perfect dominating set of  (TSPD-set) if for every vertex  is adjacent to exactly on evertex  and is a dominating set of  The minimum cardinality of a twain secure perfect dominating set of  is called the twain secure perfect domination number of and is denoted by Let  denote the family of all twain secure perfect dominating sets of  with cardinality for  Let  In this article, we derive a recursive formula for and construct . We consider the polynomial  which we refer to as the twain secure perfect domination polynomial of stars using this recursive formula. In this research, we use a recursive technique to generate all twain secure perfect dominating sets of stars and twain secure perfect domination polynomials of stars.