R. Davila, M.A. Henning
Total forcing sets and zero forcing sets in trees
Discussiones Mathematicae Graph Theory
Received 28.09.2017, Revised 13.03.2018, Accepted 23.03.2018, doi: 10.7151/dmgt.2136

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 Ft(G). We prove that if T is a tree of order n ≥ 3 with maximum degree Δ and with n1 leaves, then n1 \le Ft(T) \le \frac{1}{Δ}((Δ - 1)n + 1). In both lower and upper bounds, we characterize the infinite family of trees achieving equality. Further we show that Ft(T) ≥ F(T)+1, and we characterize the extremal trees for which equality holds.
forcing set, forcing number, total forcing set, total forcing number