A New Bound on the Total Domination Subdivision Number

Abstract

A set S of vertices of a graph G = (V, E) 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 γ _t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number $sd_{\gamma_{t}}(G)$ 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 every simple connected graph G of order n ≥ 3,

$${\rm sd}_{\gamma_{t}}(G)\le 3 +{\rm min}\{d_2(v); v\in V \, {\rm and}\, d(v)\ge 2\}$$

where d ₂(v) is the number of vertices of G at distance 2 from v.