Strong nonsplit dominating sets and strong nonsplit domination polynomial of complement of paths

Abstract

Let [Formula: see text] be a simple graph of order [Formula: see text]. The strong nonsplit domination polynomial of a graph [Formula: see text] is [Formula: see text] where [Formula: see text] is the number of strong nonsplit dominating sets of [Formula: see text] of size [Formula: see text] and [Formula: see text] is the strong nonsplit domination number of [Formula: see text]. Let [Formula: see text] be the family of strong nonsplit dominating sets of a complement of a path [Formula: see text] ([Formula: see text]) with cardinality [Formula: see text], and let [Formula: see text]. In this paper, we construct [Formula: see text], the recursive formula for [Formula: see text] and [Formula: see text], the strong nonsplit domination polynomial of [Formula: see text]. Also, we obtain some properties of the coefficients of this polynomial.