Total Domination and minimal total domination polynomial of $H-$join graphs

Abstract

Let $H$ be a connected labeled graph. In this article, we characterize all the total dominating sets and the minimal total dominating sets of $H-$join graphs. Consequently, we compute the (multivariate) total domination polynomial and the (multivariate) minimal total domination polynomial of $H-$join graphs. We also compute the total domination number of $H-$join graphs. Finally, as an illustration, we calculate the total domination polynomial and the minimal total domination polynomial of the join of graphs, multipartite graphs, $K_n-$join graphs, $K_{n_1,...,n_m}-$join graphs, and the corona product of graphs.