On two conjectures concerning total domination subdivision number in graphs

Abstract

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$\gamma _t(G)$ is the minimum cardinality of a total dominating set of G. The total domination subdivision number$\mathrm{sd}_{\gamma _t}(G)$ 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 $n\ge 3$, $\mathrm{sd}_{\gamma _t}(G)\le \gamma _t(G)+1$ and for any connected graph G of order $n\ge 5$, $\mathrm{sd}_{\gamma _t}(G)\le \frac{n+1}{2}$, answering two conjectures posed in Favaron et al. (J Comb Optim 20:76–84, 2010a).