Bootstrap percolation and $P_3$-hull number in direct products of graphs

Abstract

The $r$-neighbor bootstrap percolation is a graph infection process based on the update rule by which a vertex with $r$ infected neighbors becomes infected. We say that an initial set of infected vertices propagates if all vertices of a graph $G$ are eventually infected, and the minimum cardinality of such a set in $G$ is called the $r$-bootstrap percolation number, $m(G,r)$, of $G$. In this paper, we study percolating sets in direct products of graphs. While in general graphs there is no non-trivial upper bound on $m(G\times H,r)$, we prove several upper bounds under the assumption $δ(G)\ge r$. We also characterize the connected graphs $G$ and $H$ with minimum degree $2$ that satisfy $m(G \times H, 2) = \frac{|V(G \times H)|}{2}$. In addition, we determine the exact values of $m(P_n \times P_m, 2)$, which are $m+n-1$ if $m$ and $n$ are of different parities, and $m+n$ otherwise.

Figures (6)

• Figure 1: Graphs $G$ and $G\times K_2$ with their percolating sets.
• Figure 2: Graph $G_1$ in $P_8\times P_8$ with its percolating set.
• Figure 3: Graph $G_2$ in $P_7 \times P_6$ with a percolating set.
• Figure 4: Graph $H_1$ with a percolating set.
• Figure 5: Graph $H_2$ with a percolating set.

Theorems & Definitions (19)

• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Corollary 3.4
• Proposition 4.1
• proof
• Corollary 4.2
• Lemma 4.3
• proof