## 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). We prove that if T is a tree of order n ≥ 3 with maximum degree ∆ and with n_{1} leaves, then n_{1} ≤ F_{t}(T) ≤ _{∆}^{1} ((∆ − 1)n + 1). In both lower and upper bounds, we characterize the infinite family of trees achieving equality. Further we show that F_{t}(T) ≥ F(T) + 1, and we characterize the extremal trees for which equality holds.

Original language | English |
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Pages (from-to) | 733-754 |

Number of pages | 22 |

Journal | Discussiones Mathematicae - Graph Theory |

Volume | 40 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2020 |

## Keywords

- Forcing number
- Forcing set
- Total forcing number
- Total forcing set

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics