• ON TOTAL DOMINATION SETS AND POLYNOMIALS OF CYCLES

Abstract

Let G = (V, E) be a graph without isolated vertices. A set S ⊆ V is a total dominating set of G, if every vertex u ∈ V is adjacent to an element of S. Let 𝔇t (Cn, i) be the family of total dominating sets of a cycle Cn with cardinality i. Let dt (Cn, i) be the number of total dominating sets in 𝔇t(Cn, i). In this paper, we study the concept of total domination polynomial for any cycle Cn. The total domination polynomial for any cycle Cn is the polynomial Dt(Cn, x) = ∑_(i=1+n/2)^n▒dt (Cn, i) xi , if n ≡ 2(mod4) and Dt(Cn, x) = ∑_(i=⌈n/2⌉)^n▒dt (Cn, i) xi if n ≢ 2(mod4). We obtain some properties of Dt(Cn ,x) and its coefficients. Also, we calculate the reduction formula to derive the total domination polynomial of cycles. xi if n ≢ 2(mod4). We obtain some properties of Dt(Cn ,x) and its coefficients. Also, we calculate the reduction formula to derive the total domination polynomial of cycles.