Matching, Path Covers, and Total Forcing Sets

TL;DR

This paper investigates the total forcing number of trees, establishing bounds involving path cover and matching numbers, and characterizes trees where these bounds are tight.

Contribution

It introduces new bounds for the total forcing number of trees based on path cover and matching numbers, and characterizes extremal trees achieving these bounds.

Findings

The total forcing number of a tree satisfies $ ext{pc}(T)+1 \\le F_t(T) \\le 2 ext{pc}(T)$.

The paper proves that $F_t(T) \\le \\alpha'(T) + ext{pc}(T)$ for trees.

Extremal trees achieving equality in these bounds are characterized.

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