Dominating Sets and Domination Polynomial of Fan Related Graphs

T.M. Selvarajan and D. Ahila Jeyanthi

Let G = (V, E) be a simple graph of order | V |= m. A set S ⊆ V is a dominating set of G, if every vertex in (V − S) is adjacent to at least one vertex in S , or equivalently N[S] = V, the closed neighborhood of S. Let 𝐹𝑚𝑘 be the family of dominating sets of Fan graph 𝐹𝑚 with cardinality k. Let d( 𝐹𝑚, k) = | 𝐹𝑚𝑘| . The minimum cardinality taken over all dominating sets in 𝐹𝑚 is called domination number ( 𝐹𝑚) . In this paper, we construct 𝐹𝑚𝑘 and obtain a recursive formula for 𝑑( 𝐹𝑚,𝑘). Using this recursive formula, consider the polynomial 𝐷( 𝐹𝑚,𝑥)= 𝑑 𝐹𝑚,𝑘 𝑥𝑘,𝑚𝑘=𝛾( 𝐹𝑚) call domination polynomial of Fan graph and obtain some properties of this polynomial.

Volume 12 | Issue 3

Pages: 233-243

DOI: 10.5373/JARDCS/V12I3/20201187