Bounding the total forcing number of graphs

• Published in: Journal of combinatorial optimization Vol. 46; no. 4
• Main Authors: Ji, Shengjin, He, Mengya, Li, Guang, Pan, Yingui, Zhang, Wenqian
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.11.2023; Springer Nature B.V
• Subjects: Apexes; Color; Combinatorics; Convex and Discrete Geometry; Graph theory; Graphs; Mathematical Modeling and Industrial Mathematics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Theory of Computation; Vertex sets; Cyclomatic number; Spanning tree; Zero forcing set; Cactus graphs; 05C69; 05C57; 05C05; Total forcing set
• ISSN: 1382-6905; 1573-2886
• DOI: 10.1007/s10878-023-01089-4

In recent years, a dynamic coloring, named as zero forcing, of the vertices in a graph have attracted many researchers. For a given G and a vertex subset S , assigning each vertex of S black and each vertex of V \ S no color, if one vertex u ∈ S has a unique neighbor v in V \ S , then u forces v to color black. S is called a zero forcing set if S can be expanded to the entire vertex set V by repeating the above forcing process. S is regarded as a total forcing set if the subgraph G [ S ] satisfies δ ( G [ S ] ) ≥ 1 . The minimum cardinality of a total forcing set in G , denoted by F t ( G ) , is named the total forcing number of G . For a graph G , p ( G ), q ( G ) and ϕ ( G ) denote the number of pendant vertices, the number of vertices with degree at least 3 meanwhile having one pendant path and the cyclomatic number of G , respectively. In the paper, by means of the total forcing set of a spanning tree regarding a graph G , we verify that F t ( G ) ≤ p ( G ) + q ( G ) + 2 ϕ ( G ) . Furthermore, all graphs achieving the equality are determined.