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 $\text{pc}(G)$, 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 ${\alpha }^{\mathrm{\prime }}(T)$, is the number of edges in a maximum matching of $G$. Let $T$ be a tree of order at least two. We observe that $\text{pc}(T)+1\le F_t(T)\le 2\text{pc}(T)$, and we prove that $F_t(T) \le \alpha’(T) + \pc(T)$. Further, we characterize the extremal trees achieving equality in these bounds.