On Trees and Unicyclic Graphs with Equal Forcing-Type Numbers

作者：Jiang, Hui; Li, Wenjing^*; Zhang, Jingshu

来源：Bulletin of the Malaysian Mathematical Sciences Society, 2022, 45(4): 1607-1620.

DOI：10.1007/s40840-022-01276-2

• Forcing number
• Total forcing number
• Connected forcing number
• Tree
• Unicyclic graph

Zero forcing is an iterative graph coloring process whereby a colored vertex with a single uncolored neighbor to be colored. A vertex set S in a graph G is a forcing set if by iteratively applying the zero forcing process, all vertices of V(G) become colored. If S has the added property that it induces an isolate-free subgraph, then S is a total forcing set of G; while S is a connected forcing set of G if it induces a connected subgraph. Let F(G), F-t(G) and F-c(G) denote the forcing number, total forcing number, and connected forcing number of G, i.e., the minimum cardinality taken over all forcing, total forcing, and connected forcing sets of G, respectively. In this paper, we characterize trees and unicyclic graphs G with F(G) = F-t(G) and F-t(G) = F-c(G), respectively.

• 单位
  Yantai University