Connected ۲-Dominating Sets and Connected ۲-Dominating Polynomials in Friendship Graphs

Let \( G = (V, E) \) be a simple graph. A subset \( D \subseteq V \) is called a connected ۲-dominating set of \( G \) if every vertex in \( V \setminus D \) is adjacent to at least two vertices of \( D \), and the induced subgraph \( G[D] \) is connected. The minimum size of such a set is referred to as the connected ۲-domination number of \( G \), denoted by \( \gamma_۲^c(G) \).  In this work, we investigate the enumeration of connected ۲-dominating sets in graphs. For this purpose, we define a generating polynomial, called the connected ۲-domination polynomial, which encodes the number of such sets of different cardinalities. Furthermore, several fundamental properties of this polynomial are studied, and explicit forms are derived for certain graph families, in particular the friendship graphs \( F_n \).

کلیدواژه ها:

Connected ۲-domination ، connected ۲-domination polynomials ، friendship graphs