Total forcing sets and zero forcing sets in trees

A dynamic coloring of the vertices of a graph G starts with an initial subset S of colored vertices, with all remaining vertices being non-colored. At each discrete time interval, a colored vertex with exactly one non-colored neighbor forces this non-colored neighbor to be colored. The initial set S is called a forcing set of G if, by iteratively applying the forcing process, every vertex in G becomes colored. If the initial set S has the added property that it induces a subgraph of G without isolated vertices, then S is called a total forcing set in G. The minimum cardinality of a total forcing set in G is its total forcing number, denoted F_t(G). We prove that if T is a tree of order n ≥ 3 with maximum degree ∆ and with n₁ leaves, then n₁ ≤ F_t(T) ≤ _∆¹ ((∆ − 1)n + 1). In both lower and upper bounds, we characterize the infinite family of trees achieving equality. Further we show that F_t(T) ≥ F(T) + 1, and we characterize the extremal trees for which equality holds.