Leaky forcing and resilience of Cartesian products of $K_n$

Abstract

Zero forcing is a process on a graph $G = (V,E)$ in which a set of initially colored vertices,$B_0(G) \subset V(G)$, can color their neighbors according to the color change rule. The color change rule states that if a vertex $v$ can color a neighbor $u$ if $u$ is the only uncolored neighbor of $v$. If a vertex $v$ colors its neighbor, $u$, $v$ is said to force $u$. Leaky forcing is a recently introduced variant of zero forcing in which some vertices cannot force their neighbors, even if they satisfy the color change rule. This variation has been studied for limited families of graphs with particular structure, such as products of paths and discrete hypercubes. A concept closely related to $\ell$-leaky forcing is $\ell$-resilience. A graph is said to be $\ell$-resilient if its $\ell$-leaky forcing number equals its zero forcing number. In this paper, we prove direct products of $K_n$ with $P_t$ and $K_n$ with $C_t$ is 1-resilient and conjecture the former is not 2-resilient.

Figures (6)

• Figure 1: $B_1$ for $K_4 \times P_6$ consists of the blue vertices.
• Figure 2: $B_1$ for $K_4 \times P_7$ consists of the blue vertices. This contains $B_1$ for $K_4 \times P_6$ in the first six columns.
• Figure 3: An intuition as to why $K_n \times P_t$ is not 2-resilient. $B_1$ consists of the blue and red vertices, however the red vertices are leaky.
• Figure 4: $B_1$ for $K_5 \times P_8$ consists of the blue vertices.
• Figure 5: $K_3 \times K_3$. The blue shaded vertices form a (not unique) $1$-leaky forcing set.

Theorems & Definitions (7)

• Theorem 2.1
• proof
• Conjecture 2.2
• Theorem 2.3
• proof
• Theorem 2.5
• proof