arXiv subjects

Boštjan Brešar

Publications and source records attributed to Boštjan Brešar.

At least 37 records · Page 2Linked to original sources

arXiv2024

Claw-free cubic graphs are $(1, 1, 2, 2)$-colorable

A $(1,1,2,2)$-coloring of a graph is a partition of its vertex set into four sets two of which are independent and the other two are $2$-packings. In this paper, we prove that every claw-free cubic graph admits a $(1,1,2,2)$-coloring. This implies that the conjecture from [Packing chromatic number, $(1,1,2,2)$-colorings, and characterizing the Petersen graph, Aequationes Math.\ 91 (2017) 169--184] that the packing chromatic number of subdivisions of subcubic graphs is at most $5$ is true in the case of claw-free cubic graphs.

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arXiv2024

Isolation game on graphs

Given a graph $G$ and a family of graphs $\cal F$, an $\cal F$-isolating set, as introduced by Caro and Hansberg, is any set $S\subset V(G)$ such that $G - N[S]$ contains no member of $\cal F$ as a subgraph. In this paper, we introduce a game in which two players with opposite goals are together building an $\cal F$-isolating set in $G$. Following the domination games, Dominator (Staller) wants that the resulting $\cal F$-isolating set obtained at the end of the game, is as small (as big) as possible, which leads to the graph invariant called the game $\cal F$-isolation number, denoted $ι_{\rm g}(G,\cal F)$. We prove that the Continuation Principle holds in the $\cal F$-isolation game, and that the difference between the game $\cal F$-isolation numbers when either Dominator or Staller starts the game is at most $1$. Considering two arbitrary families of graphs $\cal F$ and $\cal F'$, we find relations between them that ensure $ι_{\rm g}(G,{\cal{F}}') \leq ι_{\rm g}(G,{\cal{F}})$ for any graph $G$. A special focus is given on the isolation game, which takes place when ${\cal F}=\{K_2\}$. We prove that $ι_{\rm g}(G,\{K_2\})\le |V(G)|/2$ for any graph $G$, and conjecture that $\lceil 3|V(G)|/7\rceil$ is the actual (sharp) upper bound. We prove that the isolation game on a forest when Dominator has the first move never lasts longer than the one in which Staller starts the game. Finally, we prove good lower and upper bounds on the game isolation numbers of paths $P_n$, which lead to the exact values $ι_{\rm g}(P_n,\{K_2\})=\left\lfloor\frac{2n+2}{5}\right\rfloor$ when $n \equiv i \pmod 5$ and $i \in \{1,2,3\}$.

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arXiv2024

Injective colorings of Sierpiński-like graphs and Kneser graphs

Two relationships between the injective chromatic number and, respectively, chromatic number and chromatic index, are proved. They are applied to determine the injective chromatic number of Sierpiński graphs and to give a short proof that Sierpiński graphs are Class $1$. Sierpiński-like graphs are also considered, including generalized Sierpiński graphs over cycles and rooted products. It is proved that the injective chromatic number of a rooted product of two graphs lies in a set of six possible values. Sierpiński graphs and Kneser graphs $K(n,r)$ are considered with respect of being perfect injectively colorable, where a graph is perfect injectively colorable if it has an injective coloring in which every color class forms an open packing of largest cardinality. In particular, all Sierpiński graphs and Kneser graphs $K(n, r)$ with $n \ge 3r-1$ are perfect injectively colorable graph, while $K(7,3)$ is not.

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arXiv2024

Bootstrap percolation in strong products of graphs

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.

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arXiv2024

$S$-packing colorings of distance graphs with distance sets of cardinality $2$

For a non-decreasing sequence $S=(s_1,s_2,\ldots)$ of positive integers, a partition of the vertex set of a graph $G$ into subsets $X_1,\ldots, X_\ell$, such that vertices in $X_i$ are pairwise at distance greater than $s_i$ for every $i\in\{1,\ldots,\ell\}$, is called an $S$-packing $\ell$-coloring of $G$. The minimum $\ell$ for which $G$ admits an $S$-packing $\ell$-coloring is called the $S$-packing chromatic number of $G$, denoted by $χ_S(G)$. In this paper, we consider $S$-packing colorings of distance graphs $G(\mathbb{Z},\{k,t\})$, where $k$ and $t$ are positive integers, which are the graphs whose vertex set is $\mathbb{Z}$, and two vertices $x,y\in \mathbb{Z}$ are adjacent whenever $|x-y|\in\{k,t\}$. We complement partial results from two earlier papers, thus determining all values of $χ_S(G(\mathbb{Z},\{k,t\}))$ when $S$ is any sequence with $s_i\le 2$ for all $i$. In particular, if $S=(1,1,2,2,\ldots)$, then the $S$-packing chromatic number is $2$ if $k+t$ is even, and $4$ otherwise, while if $S=(1,2,2,\ldots)$, then the $S$-packing chromatic number is $5$, unless $\{k,t\}=\{2,3\}$ when it is $6$; when $S=(2,2,2,\ldots)$, the corresponding formula is more complex.

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arXiv2024

Indicated domination game

Motivated by the success of domination games and by a variation of the coloring game called the indicated coloring game, we introduce a version of domination games called the indicated domination game. It is played on an arbitrary graph $G$ by two players, Dominator and Staller, where Dominator wants to finish the game in as few rounds as possible while Staller wants just the opposite. In each round, Dominator indicates a vertex $u$ of $G$ that has not been dominated by previous selections of Staller, which, by the rules of the game, forces Staller to select a vertex in the closed neighborhood of $u$. The game is finished when all vertices of $G$ become dominated by the vertices selected by Staller. Assuming that both players are playing optimally according to their goals, the number of selected vertices during the game is the indicated domination number, $γ_i(G)$, of $G$. We prove several bounds on the indicated domination number expressed in terms of other graph invariants. In particular, we find a place of the new graph invariant in the well-known domination chain, by showing that $γ_i(G)\ge Γ(G)$ for all graphs $G$, and by showing that the indicated domination number is incomparable with the game domination number and also with the upper irredundance number. In connection with the trivial upper bound $γ_i(G)\le n(G)-δ(G)$, we characterize the class of graphs $G$ attaining the bound provided that $n(G)\ge 2δ(G)+2$. We prove that in trees, split graphs and grids the indicated domination number equals the independence number. We also find a formula for the indicated domination number of powers of paths, from which we derive that there exist graphs in which the indicated domination number is arbitrarily larger than the upper irredundance number.

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arXiv2024

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

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.

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arXiv2024

Edge open packing: complexity, algorithmic aspects, and bounds

Given a graph $G$, two edges $e_{1},e_{2}\in E(G)$ are said to have a common edge $e$ if $e$ joins an endvertex of $e_{1}$ to an endvertex of $e_{2}$. A subset $B\subseteq E(G)$ is an edge open packing set in $G$ if no two edges of $B$ have a common edge in $G$, and the maximum cardinality of such a set in $G$ is called the edge open packing number, $ρ_{e}^{o}(G)$, of $G$. In this paper, we prove that the decision version of the edge open packing number is NP-complete even when restricted to graphs with universal vertices, Eulerian bipartite graphs, and planar graphs with maximum degree $4$, respectively. In contrast, we present a linear-time algorithm that computes the edge open packing number of a tree. We also resolve two problems posed in the seminal paper [Edge open packing sets in graphs, RAIRO-Oper.\ Res.\ 56 (2022) 3765--3776]. Notably, we characterize the graphs $G$ that attain the upper bound $ρ_e^o(G)\le |E(G)|/δ(G)$, and provide lower and upper bounds for the edge-deleted subgraph of a graph and establish the corresponding realization result.

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arXiv2023

$k$-Domination invariants on Kneser graphs

In this follow-up to [M.G.~Cornet, P.~Torres, arXiv:2308.15603], where the $k$-tuple domination number and the 2-packing number in Kneser graphs $K(n,r)$ were studied, we are concerned with two variations, the $k$-domination number, ${γ_{k}}(K(n,r))$, and the $k$-tuple total domination number, ${γ_{t\times k}}(K(n,r))$, of $K(n,r)$. For both invariants we prove monotonicity results by showing that ${γ_{k}}(K(n,r))\ge {γ_{k}}(K(n+1,r))$ holds for any $n\ge 2(k+r)$, and ${γ_{t\times k}}(K(n,r))\ge {γ_{t\times k}}(K(n+1,r))$ holds for any $n\ge 2r+1$. We prove that ${γ_{k}}(K(n,r))={γ_{t\times k}}(K(n,r))=k+r$ when $n\geq r(k+r)$, and that in this case every ${γ_{k}}$-set and ${γ_{t\times k}}$-set is a clique, while ${γ_{k}}(r(k+r)-1,r)={γ_{t\times k}}(r(k+r)-1,r)=k+r+1$, for any $k\ge 2$. Concerning the 2-packing number, $ρ_2(K(n,r))$, of $K(n,r)$, we prove the exact values of $ρ_2(K(3r-3,r))$ when $r\ge 10$, and give sufficient conditions for $ρ_2(K(n,r))$ to be equal to some small values by imposing bounds on $r$ with respect to $n$. We also prove a version of monotonicity for the $2$-packing number of Kneser graphs.

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arXiv2023

Computation of Grundy dominating sequences in (co-)bipartite graphs

A sequence $S$ of vertices of a graph $G$ is called a dominating sequence of $G$ if $(i)$ each vertex $v$ of $S$ dominates a vertex of $G$ that was not dominated by any of the vertices preceding vertex $v$ in $S$, and $(ii)$ every vertex of $G$ is dominated by at least one vertex of $S$. The Grundy Domination problem is to find a longest dominating sequence for a given graph $G$. It has been known that the decision version of the Grundy Domination problem is NP-complete even when restricted to chordal graphs. In this paper, we prove that the decision version of the Grundy Domination problem is NP-complete for bipartite graphs and co-bipartite graphs. On the positive side, we present a linear-time algorithm that solves the Grundy Domination problem for chain graphs, which form a subclass of bipartite graphs.

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arXiv2023

Lower (total) mutual visibility in graphs

Given a graph $G$, a set $X$ of vertices in $G$ satisfying that between every two vertices in $X$ (respectively, in $G$) there is a shortest path whose internal vertices are not in $X$ is a mutual-visibility (respectively, total mutual-visibility) set in $G$. The cardinality of a largest (total) mutual-visibility set in $G$ is known under the name (total) mutual-visibility number, and has been studied in several recent works. In this paper, we propose two lower variants of the mentioned concepts, defined as the smallest possible cardinality among all maximal (total) mutual-visibility sets in $G$, and denote them by $μ^{-}(G)$ and $μ_t^{-}(G)$, respectively. While the total mutual-visibility number is never larger than the mutual-visibility number in a graph $G$, we prove that both differences $μ^{-}(G)-μ_t^{-}(G)$ and $μ_t^{-}(G)-μ^{-}(G)$ can be arbitrarily large. We characterize graphs $G$ with some small values of $μ^{-}(G)$ and $μ_t^{-}(G)$, and prove a useful tool called Neighborhood Lemma, which enables us to find upper bounds on the lower mutual-visibility number in several classes of graphs. We compare the lower mutual-visibility number with the lower general position number, and find a close relationship with Bollobás-Wessel theorem when this number is considered in Cartesian products of complete graphs. Finally, we also prove the NP-completeness of the decision problem related to $μ_t^{-}(G)$.

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arXiv2023

Bounds on zero forcing using (upper) total domination and minimum degree

While a number of bounds are known on the zero forcing number $Z(G)$ of a graph $G$ expressed in terms of the order of a graph and maximum or minimum degree, we present two bounds that are related to the (upper) total domination number $γ_t(G)$ (resp. $Γ_t(G)$) of $G$. We prove that $Z(G)+γ_t(G)\le n(G)$ and $Z(G)+\frac{Γ_t(G)}{2}\le n(G)$ holds for any graph $G$ with no isolated vertices of order $n(G)$. Both bounds are sharp as demonstrated by several infinite families of graphs. In particular, we show that every graph $H$ is an induced subgraph of a graph $G$ with $Z(G)+\frac{Γ_t(G)}{2}=n(G)$. Furthermore, we prove a characterization of graphs with power domination equal to $1$, from which we derive a characterization of the extremal graphs attaining the trivial lower bound $Z(G)\ge δ(G)$. The class of graphs that appears in the corresponding characterizations is obtained by extending an idea from [D.D.~Row, A technique for computing the zero forcing number of a graph with a cut-vertex, Linear Alg.\ Appl.\ 436 (2012) 4423--4432], where the graphs with zero forcing number equal to $2$ were characterized.

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arXiv2023

Spreading in graphs

Several concepts that model processes of spreading (of information, disease, objects, etc.) in graphs or networks have been studied. In many contexts, we assume that some vertices of a graph $G$ are contaminated initially, before the process starts. By the $q$-forcing rule, a contaminated vertex having at most $q$ uncontaminated neighbors enforces all the neighbors to become contaminated, while by the $p$-percolation rule, an uncontaminated vertex becomes contaminated if at least $p$ of its neighbors are contaminated. In this paper, we consider sets $S$ that are at the same time $q$-forcing sets and $p$-percolating sets, and call them $(p,q)$-spreading sets. Given positive integers $p$ and $q$, the minimum cardinality of a $(p,q)$-spreading set in $G$ is a $(p,q)$-spreading number, $σ_{(p,q)}(G)$, of $G$. While $q$-forcing sets have been studied in a dozen of papers, the decision version of the corresponding graph invariant has not been considered earlier, and we fill the gap by proving its NP-completeness. This, in turn, enables us to prove the NP-completeness of the decision version of the $(p,q)$-spreading number in graphs for an arbitrary choice of $p$ and $q$. Again, for every $p\in \mathbb{N}$ and $q\in\mathbb{N}\cup\{\infty\}$, we find a linear-time algorithm for determining the $(p,q)$-spreading number of a tree. In addition, we present a lower and an upper bound on the $(p,q)$-spreading number of a tree and characterize extremal families of trees. In the case of square grids, we combine some known results and new results on $(2,1)$-spreading and $(4,q)$-spreading to obtain $σ_{(p,q)}(P_m\Box P_n)$ for all $(p,q)\in (\mathbb{N}\setminus\{3\})\times (\mathbb{N}\cup\{\infty\})$ and all $m,n\in\mathbb{N}$.

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arXiv2023

Packings in bipartite prisms and hypercubes

The $2$-packing number $ρ_2(G)$ of a graph $G$ is the cardinality of a largest $2$-packing of $G$ and the open packing number $ρ^{\rm o}(G)$ is the cardinality of a largest open packing of $G$, where an open packing (resp. $2$-packing) is a set of vertices in $G$ no two (closed) neighborhoods of which intersect. It is proved that if $G$ is bipartite, then $ρ^{\rm o}(G\Box K_2) = 2ρ_2(G)$. For hypercubes, the lower bounds $ρ_2(Q_n) \ge 2^{n - \lfloor \log n\rfloor -1}$ and $ρ^{\rm o}(Q_n) \ge 2^{n - \lfloor \log (n-1)\rfloor -1}$ are established. These findings are applied to injective colorings of hypercubes. In particular, it is demonstrated that $Q_9$ is the smallest hypercube which is not perfect injectively colorable. It is also proved that $γ_t(Q_{2^k}\times H) = 2^{2^k-k}γ_t(H)$, where $H$ is an arbitrary graph with no isolated vertices.

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arXiv2023

Graphs with equal Grundy domination and independence number

The Grundy domination number, ${γ_{\rm gr}}(G)$, of a graph $G$ is the maximum length of a sequence $(v_1,v_2,\ldots, v_k)$ of vertices in $G$ such that for every $i\in \{2,\ldots, k\}$, the closed neighborhood $N[v_i]$ contains a vertex that does not belong to any closed neighborhood $N[v_j]$, where $j<i$. It is well known that the Grundy domination number of any graph $G$ is greater than or equal to the upper domination number $Γ(G)$, which is in turn greater than or equal to the independence number $α(G)$. In this paper, we initiate the study of the class of graphs $G$ with $Γ(G)={γ_{\rm gr}}(G)$ and its subclass consisting of graphs $G$ with $α(G)={γ_{\rm gr}}(G)$. We characterize the latter class of graphs among all twin-free connected graphs, provide a number of properties of these graphs, and prove that the hypercubes are members of this class. In addition, we give several necessary conditions for graphs $G$ with $Γ(G)={γ_{\rm gr}}(G)$ and present large families of such graphs.

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arXiv2022

Orientable domination in product-like graphs

The orientable domination number, ${\rm DOM}(G)$, of a graph $G$ is the largest domination number over all orientations of $G$. In this paper, ${\rm DOM}$ is studied on different product graphs and related graph operations. The orientable domination number of arbitrary corona products is determined, while sharp lower and upper bounds are proved for Cartesian and lexicographic products. A result of Chartrand et al. from 1996 is extended by establishing the values of ${\rm DOM}(K_{n_1,n_2,n_3})$ for arbitrary positive integers $n_1,n_2$ and $n_3$. While considering the orientable domination number of lexicographic product graphs, we answer in the negative a question concerning domination and packing numbers in acyclic digraphs posed in [Domination in digraphs and their direct and Cartesian products, J. Graph Theory 99 (2022) 359-377].

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arXiv2022

Injective coloring of graphs revisited

An open packing in a graph $G$ is a set $S$ of vertices in $G$ such that no two vertices in $S$ have a common neighbor in $G$. The injective chromatic number $χ_i(G)$ of $G$ is the smallest number of colors assigned to vertices of $G$ such that each color class is an open packing. Alternatively, the injective chromatic number of $G$ is the chromatic number of the two-step graph of $G$, which is the graph with the same vertex set as $G$ in which two vertices are adjacent if they have a common neighbor. The concept of injective coloring has been studied by many authors, while in the present paper we approach it from two novel perspectives, related to open packings and the two-step graph operation. We prove several general bounds on the injective chromatic number expressed in terms of the open packing number. In particular, we prove that $χ_{i}(G)\geq \frac{1}{2}+\sqrt{\frac{1}{4}+\frac{2m-n}{\opack}}$ holds for any connected graph $G$ of order $n\geq2$, size $m$, and the open packing number $\opack$, and characterize the class of graphs attaining the bound. Regarding the well-known bound $χ_i(G)\ge Δ(G)$, we describe the family of extremal graphs and prove that deciding when the equality holds (even for regular graphs) is NP-complete, solving an open problem from an earlier paper. Next, we consider the chromatic number of the two-step graph of a graph, and compare it with the clique number and the maximum degree of the graph. We present two large families of graphs in which $χ_i(G)$ equals the cardinality of a largest clique of the two-step graph of $G$. Finally, we consider classes of graphs that admit an injective coloring in which all color classes are maximal open packings. We give characterizations of three subclasses of these graphs among graphs with diameter $2$, and find a partial characterization of hypercubes with this property.

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arXiv2021

On the chromatic edge stability index of graphs

Given a non-trivial graph $G$, the minimum cardinality of a set of edges $F$ in $G$ such that $χ'(G \setminus F)<χ'(G)$ is called the chromatic edge stability index of $G$, denoted by $es_{χ'}(G)$, and such a (smallest) set $F$ is called a (minimum) mitigating set. While $1\le es_{χ'}(G)\le \lfloor n/2\rfloor$ holds for any graph $G$, we investigate the graphs with extremal and near-extremal values of $es_{χ'}(G)$. The graphs $G$ with $es_{χ'}(G)=\lfloor n/2\rfloor$ are classified, and the graphs $G$ with $es_{χ'}(G)=\lfloor n/2\rfloor-1$ and $χ'(G)=Δ(G)+1$ are characterized. We establish that the odd cycles and $K_2$ are exactly the regular connected graphs with the chromatic edge stability index $1$; on the other hand, we prove that it is NP-hard to verify whether a graph $G$ has $es_{χ'}(G)=1$. We also prove that every minimum mitigating set of an $r$-regular graph $G$, where $r\ne 4$, with $es_{χ'}(G)=2$ is a matching. Furthermore, we propose a conjecture that for every graph $G$ there exists a minimum mitigating set, which is a matching, and prove that the conjecture holds for graphs $G$ with $es_{χ'}(G)\in\{1,2,\lfloor n/2\rfloor-1,\lfloor n/2\rfloor\}$, and for bipartite graphs.

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