## Abstract

A total forcing set in a graph G is a forcing set (zero forcing set) in G which induces a subgraph without isolated vertices. Total forcing sets were introduced and first studied by Davila (2015). The total forcing number of G, denoted F _{t} (G) is the minimum cardinality of a total forcing set in G. We study basic properties of F _{t} (G), relate F _{t} (G) to various domination parameters, and establish NP-completeness of the associated decision problem for F _{t} (G). Our main contribution is to prove that if G is a connected graph of order n≥3 with maximum degree Δ, then F _{t} (G)≤([Formula presented])n, with equality if and only if G is a complete graph K _{Δ+1} , or a star K _{1,Δ} .

Original language | English |
---|---|

Pages (from-to) | 115-127 |

Number of pages | 13 |

Journal | Discrete Applied Mathematics |

Volume | 257 |

DOIs | |

Publication status | Published - 31 Mar 2019 |

## Keywords

- Dominating sets
- Forcing sets
- Total forcing number
- Total forcing sets

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics