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Povzetek: The isolation game is played on a graph $G$ by two players who take turns playing a vertex such that if $X$ is the set of already played vertices, then a vertex can be selected only if it dominates a vertex from a nontrivial component of $G \setminus N_G[X]$, where $N_G[X]$ is the set of vertices in $X$ or adjacent to a vertex in $X$. Dominator wishes to finish the game with the minimum number of played vertices, while Staller has the opposite goal. The game isolation number $\iota_{\rm g}(G)$ is the number of moves in the Dominator-start game where both players play optimally. If Staller starts the game the invariant is denoted by $\iota_{\rm g}'(G)$. In this paper, $\iota_{\rm g}(C_n)$, $\iota_{\rm g}(P_n)$, $\iota_{\rm g}'(C_n)$, and $\iota_{\rm g}'(P_n)$ are determined for all $n$. It is proved that there are only two graphs that attain equality in the upper bound $\iota_{\rm g}(G) \le \frac{1}{2}|V(G)|$, and that there are precisely eleven graphs which attain equality in the upper bound $\iota_{\rm g}'(G) \le \frac{1}{2}|V(G)|$. For trees $T$ of order at least three it is proved that $\iota_{\rm g}(T) \le \frac{5}{11}|V(T)|$. A new infinite family of graphs $G$ is also constructed for which $\iota_{\rm g}(G) = \iota_{\rm g}'(G) = \frac{3}{7}|V(G)|$ holds.
Ključne besede: isolating set, isolation game, paths and cycles, trees
Objavljeno v DiRROS: 12.06.2026; Ogledov: 115; Prenosov: 94
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Povzetek: 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.
Ključne besede: bootstrap percolation, zero forcing set, k-forcing set, spreading
Objavljeno v DiRROS: 23.09.2025; Ogledov: 564; Prenosov: 301
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Povzetek: Let $G$ and $H$ be graphs and let $f \colon V(G)\rightarrow V(H)$ be a function. The Sierpiński product of $G$ and $H$ with respect to $f$, denoted by $G \otimes _f H$, is defined as the graph on the vertex set $V(G)\times V(H)$, consisting of $|V(G)|$ copies of $H$; for every edge $gg'$ of $G$ there is an edge between copies $gH$ and $g'H$ of $H$ associated with the vertices $g$ and $g'$ of $G$, respectively, of the form $(g,f(g'))(g',f(g))$. In this paper, we define the Sierpiński domination number as the minimum of $\gamma(G\otimes _f H)$ over all functions $f \colon V(G)\rightarrow V(H)$. The upper Sierpiński domination number is defined analogously as the corresponding maximum. After establishing general upper and lower bounds, we determine the upper Sierpiński domination number of the Sierpiński product of two cycles, and determine the lower Sierpiński domination number of the Sierpiński product of two cycles in half of the cases and in the other half cases restrict it to two values.
Ključne besede: Sierpiński graph, Sierpiński product, domination number, Sierpiński domination number
Objavljeno v DiRROS: 24.01.2025; Ogledov: 899; Prenosov: 609
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Povzetek: Let $G$ and $H$ be graphs and let $f \colon V(G)\rightarrow V(H)$ be a function. The Sierpiński product of $G$ and $H$ with respect to $f$, denoted by $G \otimes _f H$, is defined as the graph on the vertex set $V(G)\times V(H)$, consisting of $|V(G)|$ copies of $H$; for every edge $gg'$ of $G$ there is an edge between copies $gH$ and $g'H$ of $H$ associated with the vertices $g$ and $g'$ of $G$, respectively, of the form $(g,f(g'))(g',f(g))$. The Sierpiński metric dimension and the upper Sierpiński metric dimension of two graphs are determined. Closed formulas are determined for Sierpiński products of trees, and for Sierpiński products of two cycles where the second factor is a triangle. We also prove that the layers with respect to the second factor in a Sierpiński product graph are convex.
Ključne besede: Sierpiński product of graphs, metric dimension, trees, convex subgraph
Objavljeno v DiRROS: 16.02.2024; Ogledov: 1509; Prenosov: 934
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