Total forcing sets and zero forcing sets in trees

Randy Davila, Michael A. Henning

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)


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 Ft(G). We prove that if T is a tree of order n ≥ 3 with maximum degree ∆ and with n1 leaves, then n1 ≤ Ft(T) ≤ 1 ((∆ − 1)n + 1). In both lower and upper bounds, we characterize the infinite family of trees achieving equality. Further we show that Ft(T) ≥ F(T) + 1, and we characterize the extremal trees for which equality holds.

Original languageEnglish
Pages (from-to)733-754
Number of pages22
JournalDiscussiones Mathematicae - Graph Theory
Issue number3
Publication statusPublished - 2020


  • Forcing number
  • Forcing set
  • Total forcing number
  • Total forcing set

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics


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