## 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 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.

Original language | English |
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Pages (from-to) | 131-147 |

Number of pages | 17 |

Journal | Quaestiones Mathematicae |

Volume | 43 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2 Jan 2020 |

## Keywords

- Matching
- path cover
- total forcing set

## ASJC Scopus subject areas

- Mathematics (miscellaneous)