Matching, path covers, and total forcing sets

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). The path cover number of G, denoted pc(G), is the minimum number of vertex disjoint paths such that every vertex belongs to a path in the cover, while the matching number of G, denoted α^′(G), is the number of edges in a maximum matching of G. Let T be a tree of order at least two. We observe that pc(T) + 1 ≤ F_t(T) ≤ 2pc(T), and we prove that F_t(T) ≤ α^′(T) + pc(T). Further, we characterize the extremal trees achieving equality in these bounds.