On the total forcing number of a graph

A total forcing set in a graph G is a forcing set (zero forcing set) in G which induces a subgraph without isolated vertices. Total forcing sets were introduced and first studied by Davila (2015). The total forcing number of G, denoted F _t (G) is the minimum cardinality of a total forcing set in G. We study basic properties of F _t (G), relate F _t (G) to various domination parameters, and establish NP-completeness of the associated decision problem for F _t (G). Our main contribution is to prove that if G is a connected graph of order n≥3 with maximum degree Δ, then F _t (G)≤([Formula presented])n, with equality if and only if G is a complete graph K _Δ+1 , or a star K _1,Δ .