Zero Forcing of Generalized Hierarchical Products of Graphs

Abstract

Zero forcing is a graph propagation process for which vertices fill-in (or propagate information to) neighbor vertices if all neighbors except for one, are filled. The zero-forcing number is the smallest number of vertices that must be filled to begin the process so that the entire graph or network becomes filled. In this paper, bounds are provided on the zero forcing number of generalized hierarchical products.

Figures (13)

• Figure 1: $G =(\{1, 2, 3, 4\}$, $\{12$, $23$, $24$, $34$})
• Figure 2: $Z(G)=2$; $pt(G,S)=2$
• Figure 3: $G = P_5$
• Figure 4: $G = C_5$
• Figure 5: $G = K_5$

Theorems & Definitions (29)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Proposition 4.1
• Theorem 5.1
• proof
• Theorem 5.2