Domination and minimal domination polynomial of H-generalized join graphs

Let H be a connected labeled graph. A H -generalized join graph is a graph obtained by H -generalized join operation of family of graphs G = \{G_v : v ∈ V(H)\} constrained by family of vertex subsets S = \{S_v ⊆ V(G_v) : v ∈ 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 Kn_1, ..., n_m -generalized join graphs.