Domination and minimal domination polynomial of $H-$generalized join graphs

Abstract

Let $H$ be a connected labeled graph. A  \emph{$H-$generalized  join graph} is a graph obtained by $H-$generalized join operation of family of graphs $\mathcal G=\{G_v:v\in V(H)\}$ constrained by family of vertex subsets $\mathcal{S}=\{S_v\subset V(G_v):v\in V(H)\}$.
In this article, we characterize all the dominating sets and the minimal dominating sets of  $H-$generalized  join graphs. Consequently, we compute the
(multivariate) domination polynomial and the minimal domination polynomial of $H-$generalized  join graphs. We also compute the domination number of  $H-$generalized  join graphs. Finally, as an illustration, we calculate the domination polynomial and the minimal domination polynomial of multipartite graphs, the corona product of graphs, $K_n-$generalized join graphs, and $K_{n_1,...,n_m}-$generalized join graphs.