LAUSR

A subset S of vertices of a graph G without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number$$\gamma _t(G)$$γt(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number$$\mathrm{sd}_{\gamma _t}(G)$$sdγt(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. In this paper we prove that for any connected graph G of order $$n\ge 3$$n≥3, $$\mathrm{sd}_{\gamma _t}(G)\le \gamma _t(G)+1$$sdγt(G)≤γt(G)+1 and for any connected graph G of order $$n\ge 5$$n≥5, $$\mathrm{sd}_{\gamma _t}(G)\le \frac{n+1}{2}$$sdγt(G)≤n+12, answering two conjectures posed in Favaron et al. (J Comb Optim 20:76–84, 2010a).