2-Dominating sets and 2-domination polynomials of pan graph \(P_{m, 2}\)

Abstract

Let \(G\) be a simple graph of order $\mathrm{m}$. Let $D_2(G, x)$ be the family of 2-dominating sets in $G$ with size 1 . The polynomial $D_2(G, x)=\sum_{i=\gamma_2(G)}^m d_2(G, i) x^i$ is called the 2-domination polynomial of $G$. Let $D_2\left(P_{m, 2}, i\right)$ be the family of 2-dominating sets of the pan graph $P_{m, 2}$ with cardinality $i$ and let $d_2\left(P_{m, 2}, i\right)=\left|D_2\left(P_{m, 2}, i\right)\right|$. Then, the 2-domination polynomial $D_2\left(P_{m, 2}, x\right)$ of $P_{m, 2}$ is defined as, $D_2\left(P_{m, 2}, x\right)=\sum_{i=\gamma_2\left(P_{m, 2}\right)}^{m+2} d_2\left(P_{m, 2}, i\right) x^i$, where $\gamma_2\left(P_{m, 2}\right)$ is the 2 - domination number of $P_{m, 2}$. In this paper we obtain a recursive formula for $d_2\left(P_{m, 2}, i\right)$. Using this recursive formula we construct the 2-domination polynomial, $\left.D_2\left(P_{m, 2}, x\right)=\sum_{i=\left[\frac{m+2}{2}\right.}^{m+2}\right]_2\left(P_{m, 2}, i\right) x^i$, where $d_2\left(P_{m, 2}, i\right)$ is the number of 2-dominating sets of $P_{m, 2}$ of cardinality $i$ and some properties of this polynomial have been studied.