γ-total dominating graphs of lollipop, umbrella, and coconut graphs

A total dominating set of a graph $g$ is a set $dsubseteq v(g)$ such that every vertex of $g$ is adjacent to some vertex in $d$. the total domination number $gamma_{t}(g)$ of $g$ is the minimum cardinality of a total dominating set. the $gamma$-total dominating graph $td_{gamma}(g)$ of $g$ is the graph whose vertices are minimum total dominating sets, and two minimum total dominating sets of $td_{gamma}(g)$ are adjacent if they differ by only one vertex. in this paper, we determine the total domination numbers of lollipop graphs, umbrella graphs, and coconut graphs, and especially their $gamma$-total dominating graphs.