Total Forcing Sets and Zero Forcing Sets in Trees

Abstract

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≤Ft(T)≤1Δ((Δ-1)N+1){n_1} \le {F_t}\left( T \right) \le {1 \over \Delta }\left( {\left( {\Delta - 1} \right)N + 1} \right). 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.

Keywords

• forcing set
• forcing number
• total forcing set
• total forcing number
• 05c69