Note on Forcing Problem of Trees

Zero forcing (zero forcing number, denoted by F ( G )) of a connected graph G , which proposed by the AIM-minimum rank group in 2008, has been received much attention. In 2015, Davila introduced total forcing which is as a strengthening form of zero forcing. The total forcing number of G is the minimum cardinality of a total forcing set in G , denoted by F_t(G) . It is easy to deduce that for any graph G , F(G)≤ F_t(G)≤ 2F(G) . In addition, we know that F_t(T)≥ F(T)+1 for a tree T . Our interest is to study the structure of the tree T with F_t(T)= 2F(T) . In the paper, we characterize all trees meet the condition. In fact, these trees can be reconstructed by beginning with a path through repeating a tree operation.