Geodesic
Total Forcing Sets and Zero Forcing Sets in Trees
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 3, pp. 733-754

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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 Δ and with n_1 leaves, then n_1≤F_t(T)≤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 F_t(T) ≥ F (T) + 1, and we characterize the extremal trees for which equality holds.
Keywords: forcing set, forcing number, total forcing set, total forcing number 
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Davila, Randy; Henning, Michael A. Total Forcing Sets and Zero Forcing Sets in Trees. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 3, pp. 733-754. http://geodesic.mathdoc.fr/item/DMGT_2020_40_3_a2/