1. Bootstrap percolation and $P_3$-hull number in direct products of graphsBoštjan Brešar, Jaka Hedžet, Rebekah Herrman, 2026, original scientific article 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 $\delta(G)\ge r$. 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.
Keywords: bootstrap percolation, direct product of graphs, $P_3$-convexity
Published in DiRROS: 16.01.2026; Views: 283; Downloads: 204
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2. Spreading in claw-free cubic graphsBoštjan Brešar, Jaka Hedžet, Michael A. Henning, 2025, original scientific article Abstract: Let $p\in\mathbb{N}$ and $q\in\mathbb{N}\cup\{\infty\}$. We study a dynamic coloring of the vertices of a graph $G$ that starts with an initial subset $S$ of blue vertices, with all remaining vertices colored white. If a white vertex $v$ has at least $p$ blue neighbors and at least one of these blue neighbors of $v$ has at most $q$ white neighbors, then by the spreading color change rule the vertex $v$ is recolored blue. The initial set $S$ of blue vertices is a $(p,q)$-spreading set for $G$ if by repeatedly applying the spreading color change rule all the vertices of $G$ are eventually colored blue. The $(p,q)$-spreading set is a generalization of the well-studied concepts of $k$-forcing and $r$-percolating sets in graphs. For $q\ge2$, a $(1,q)$-spreading set is exactly a $q$-forcing set, and the $(1,1)$-spreading set is a $1$-forcing set (also called a zero forcing set), while for $q=\infty$, a $(p,\infty)$-spreading set is exactly a $p$-percolating set. The $(p,q)$-spreading number, $\sigma_{(p,q)}(G)$, of $G$ is the minimum cardinality of a $(p,q)$-spreading set. In this paper, we study $(p,q)$-spreading in claw-free cubic graphs. While the zero-forcing number of claw-free cubic graphs was studied earlier, for each pair of values $p$ and $q$ that are not both $1$ we either determine the $(p,q)$-spreading number of a claw-free cubic graph $G$ or show that $\sigma_{(p,q)}(G)$ attains one of two possible values.
Keywords: bootstrap percolation, zero forcing set, k-forcing set, spreading
Published in DiRROS: 23.09.2025; Views: 466; Downloads: 248
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3. Bootstrap percolation in strong products of graphsBoštjan Brešar, Jaka Hedžet, 2024, original scientific article Abstract: Given a graph $G$ and assuming that some vertices of $G$ are infected, the $r$-neighbor bootstrap percolation rule makes an uninfected vertex $v$ infected if $v$ has at least $r$ infected neighbors. The $r$-percolation number, $m(G,r)$, of $G$ is the minimum cardinality of a set of initially infected vertices in $G$ such that after continuously performing the $r$-neighbor bootstrap percolation rule each vertex of $G$ eventually becomes infected. In this paper, we consider percolation numbers of strong products of graphs. If $G$ is the strong product $G_1\boxtimes \cdots \boxtimes G_k$ of $k$ connected graphs, we prove that $m(G,r)=r$ as soon as $r\le 2^{k-1}$ and $|V(G)|\ge r$. As a dichotomy, we present a family of strong products of $k$ connected graphs with the $(2^{k-1}+1)$-percolation number arbitrarily large. We refine these results for strong products of graphs in which at least two factors have at least three vertices. In addition, when all factors $G_i$ have at least three vertices we prove that $m(G_1 \boxtimes \dots \boxtimes G_k,r)\leq 3^{k-1} -k$ for all $r\leq 2^k-1$, and we again get a dichotomy, since there exist families of strong products of $k$ graphs such that their $2^{k}$-percolation numbers are arbitrarily large. While $m(G\boxtimes H,3)=3$ if both $G$ and $H$ have at least three vertices, we also characterize the strong prisms $G\boxtimes K_2$ for which this equality holds. Some of the results naturally extend to infinite graphs, and we briefly consider percolation numbers of strong products of two-way infinite paths.
Keywords: bootstrap percolation, strong product of graphs, infinite path
Published in DiRROS: 20.11.2024; Views: 934; Downloads: 542
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4. New features in the dendroTools R package : bootstrapped and partial correlation coefficients for monthly and daily climate dataJernej Jevšenak, 2020, original scientific article Abstract: Climate-growth relationships are usually analysed using monthly climate data. The dendroTools R package also provides methodological approaches that enable climate-growth analysis for daily climate data. Such analysis reveals more complete climate signal patterns. In this article, new functions of the dendroTools R package are presented. Partial correlation coefficients are now implemented and can be used to calculate the strength of a linear relationship between two variables, while controlling for a third variable. Bootstrapped correlations can then be used to provide insights into the confidence intervals of statistical estimates. The calculation of partial and bootstrapped correlations is available for daily and monthly data. Finally, data transformation, S3 generic plotting and summary functions are also presented here.
Keywords: dendroTools, daily climate data, partial correlations, bootstrap, dendroclimatology
Published in DiRROS: 24.09.2020; Views: 2357; Downloads: 1392
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