2-Edge dominating sets and 2-Edge domination polynomials of paths

Abstract

Let $P_n$ be the path with $n$ vertices and $(n-1)$ edges. Let $D_{2 e}(G, i)$ be the family of 2-edge dominating sets in $G$ with cardinality i. The polynomial $D_{2 e}(G, i)=\Sigma_{i=\eta_e(G)}^{|E(G)|} d_{2 e}(G, i) x^i$ is called the 2-edge domination polynomial of G. In this paper, we obtain a recursive formula for $d_{2 e}\left(P_n, i\right)$. Using this recursive formula we construct 2-edge domination polynomial, $D_{2 e}\left(P_n, x\right)=\Sigma_{i=\left[\frac{n}{2}\right]}^{n-1} d_{2 e}\left(P_n, i\right) x^i$ where $d_{2 e}\left(P_n, i\right)$ is the number of 2- edge dominating sets of $P_n$ of cardinality $i$ and obtain some properties of this polynomial.