  A subset of vertices in a graph is called a total dominating set if every
vertex of the graph is adjacent to at least one vertex of this set. A total
dominating set is called minimal if it does not properly contain another total
dominating set. In this paper, we study graphs whose all minimal total
dominating sets have the same size, referred to as well-totally-dominated (WTD)
graphs. We first show that WTD graphs with bounded total domination number can
be recognized in polynomial time. Then we focus on WTD graphs with total
domination number two. In this case, we characterize triangle-free WTD graphs
and WTD graphs with packing number two, and we show that there are only
finitely many planar WTD graphs with minimum degree at least three. Lastly, we
show that if the minimum degree is at least three then the girth of a WTD graph
is at most 12. We conclude with several open questions.
