Bootstrap percolation and $P_3$-hull number in direct products of graphsBrešar, Boštjan (Avtor) Hedžet, Jaka (Avtor) Herrman, Rebekah (Avtor) bootstrap percolationdirect product of graphs$P_3$-convexityThe $r$-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 $\delta(G)\ge r$. 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. 20262026-01-16 10:38:54Neznano25333UDK: 519.17:519.2ISSN pri članku: 1234-3099DOI: 10.7151/dmgt.2603COBISS_ID: 264957187sl