twain secure perfect dominating sets and twain secure perfect domination polynomials of cycles

Let g = (v,e) be a simple graph. a set s ⊆ v is a dominating set of g, if for every vertex in vs is adjacent to at least one vertex in s. a subset s of v is called a twain secure perfect dominating set of g (tspd-set) if for every vertex v ∈ v s is adjacent to exactly one vertex u ∈ s and (s{u})∪{v} is a dominating set of g. the minimum cardinality of a twain secure perfect dominating set of g is called the twain secure perfect domination number of g and is denoted by γtsp(g).let dtsp(cn, i) denote the family of all twain secure perfect dominating sets of cn with cardinality i, for γtsp(cn)≤ i≤ n. let dtsp(cn, i) = |dtsp(cn, i)|. in this article, we derive a recursive formula for dtsp(cn, i) and construct dtsp(cn, i). weconsider the polynomial dtsp(cn, x) = σn i=γtsp(cn) dtsp(cn, i)xi, which we refer to as the twain secure perfect domination polynomial of cycles using this recursive formula. in this research, we use a recursive technique to generate all twain secure perfect dominating sets of cycles and twain secure perfect domination polynomials of cycles.