arXiv subjects

Sandi Klavžar

Publications and source records attributed to Sandi Klavžar.

At least 37 records · Page 2Linked to original sources

arXiv2025

On decycling and forest numbers of Cartesian products of trees

The decycling number $\nabla(G)$ of a graph $G$ is the minimum number of vertices that must be removed to eliminate all cycles in $G$. The forest number $f(G)$ is the maximum number of vertices that induce a forest in $G$. So $\nabla(G) + f(G) = |V(G)|$. For the Cartesian product $T \,\square\, T'$ of trees $T$ and $T'$ it is proved that $\nabla(S_n \,\square\, S_{n'}) \leq \nabla(T \,\square\, T')$, thus resolving the conjecture of Wang and Wu asserting that $f(T \,\square\, T') \leq f(S_n \,\square\, S_{n'})$. It is shown that $\nabla(T \,\square\, T') \ge |V(T)| - 1$ and the equality cases characterized. For prisms over trees, it is proved that $\nabla(T\,\square\, K_2) = α'(T)$, and for arbitrary graphs $G_1$ and $G_2$, it is proved that $\nabla(G_1 \,\square\, G_2) \geq α'(G_1) α'(G_2)$, where $α'$ is the matching number.

math.CO↗

arXiv2025

The $d$-distance $p$-packing domination number: complexity, cycles, and trees

A set of vertices $X\subseteq V(G)$ is a $d$-distance dominating set if for every $u\in V(G)\setminus X$ there exists $x\in X$ such that $d(u,x) \le d$, and $X$ is a $p$-packing if $d(u,v) \ge p+1$ for every different $u,v\in X$. The $d$-distance $p$-packing domination number $γ_d^p(G)$ of $G$ is the minimum size of a set of vertices of $G$ which is both a $d$-distance dominating set and a $p$-packing. It is proved that for every two fixed integers $d$ and $p$ with $2 \le d$ and $0 \le p \leq 2d-1$, the decision problem whether $γ_d^p(G) \leq k$ holds is NP-complete for bipartite planar graphs. A necessary and sufficient condition for the existence of a $d$-distance $p$-packing dominating set in $C_n$ is obtained and $γ_d^p(C_n)$ determined for every $d$, $p$, and $n$. For a tree $T$ on $n$ vertices with $\ell$ leaves and $s$ support vertices it is proved that (i) $γ_2^0(T) \geq \frac{n-\ell-s+4}{5}$, (ii) $\left \lceil \frac{n-\ell-s+4}{5} \right \rceil \leq γ_2^2(T) \leq \left \lfloor \frac{n+3s-1}{5} \right \rfloor$, and if $d \geq 2$, then (iii) $γ_d^2(T) \leq \frac{n-2\sqrt{n}+d+1}{d}$. Inequality (i) improves an earlier bound due to Meierling and Volkmann, and independently Raczek, Lemańska, and Cyman, while (iii) extends an earlier result for $γ_2^2(T)$ due to Henning. Sharpness of the bounds are discussed and established in most cases. It is also proved that every connected graph $G$ contains a spanning tree $T$ such that $γ_2^2(T) \leq γ_2^2(G)$.

math.CO↗

arXiv2025

Revisiting $d$-distance (independent) domination in trees and in bipartite graphs

The $d$-distance $p$-packing domination number $γ_d^p(G)$ of $G$ is the minimum size of a set of vertices of $G$ which is both a $d$-distance dominating set and a $p$-packing. In 1994, Beineke and Henning conjectured that if $d\ge 1$ and $T$ is a tree of order $n \geq d+1$, then $γ_d^1(T) \leq \frac{n}{d+1}$. They supported the conjecture by proving it for $d\in \{1,2,3\}$. In this paper, it is proved that $γ_d^1(G) \leq \frac{n}{d+1}$ holds for any bipartite graph $G$ of order $n \geq d+1$, and any $d\ge 1$. Trees $T$ for which $γ_d^1(T) = \frac{n}{d+1}$ holds are characterized. It is also proved that if $T$ has $\ell$ leaves, then $γ_d^1(T) \leq \frac{n-\ell}{d}$ (provided that $n-\ell \geq d$), and $γ_d^1(T) \leq \frac{n+\ell}{d+2}$ (provided that $n\geq d$). The latter result extends Favaron's theorem from 1992 asserting that $γ_1^1(T) \leq \frac{n+\ell}{3}$. In both cases, trees that attain the equality are characterized and relevant conclusions for the $d$-distance domination number of trees derived.

math.CO↗

arXiv2025

Maker-Breaker total domination number

The Maker-Breaker total domination number, $γ_{\rm MBT}(G)$, of a graph $G$ is introduced as the minimum number of moves of Dominator to win the Maker-Breaker total domination game, provided that he has a winning strategy and is the first to play. The Staller-start Maker-Breaker total domination number, $γ_{\rm MBT}'(G)$, is defined analogously for the game in which Staller starts. Upper and lower bounds on $γ_{\rm MBT}(G)$ and on $γ_{\rm MBT}'(G)$ are provided and demonstrated to be sharp. It is proved that for any pair of integers $(k,\ell)$ with $2\leq k\leq \ell$, (i) there exists a connected graph $G$ with $γ_{\rm MB}(G)=k$ and $γ_{\rm MBT}(G)=\ell$, (ii) there exists a connected graph $G'$ with $γ_{\rm MB}'(G')=k$ and $γ_{\rm MBT}'(G')=\ell$, and (iii) there there exists a connected graph $G''$ with $γ_{\rm MBT}(G'')=k$ and $γ_{\rm MBT}'(G'')=\ell$. Here, $γ_{\rm MB}$ and $γ_{\rm MB}'$ are corresponding invariants for the Maker-Breaker domination game.

math.CO↗

arXiv2025

Intertwining local (adjacency) metric dimension with the clique number of a graph

Let $G$ be a simple connected graph with order $ n(G)$, local metric dimension $ {\rm dim}_l(G)$, local adjacency metric dimension $ {\rm dim}_{A,l}(G)$, and clique number $ ω(G)$, where $G\not\cong K_{n(G)}$ and $ω(G)\geq3$. It is proved that $ {\rm dim}_{A,l}(G) \leq \left\lfloor \left(\frac{ω(G) - 2}{ω(G) - 1}\right)n(G)\right\rfloor$. Consequently, the conjecture asserting that the latter expression is an upper bound for ${\rm dim}_l(G)$ is confirmed. It is important to note that there are infinitely many graphs that satisfy the equalities.

math.CO↗

arXiv2025

Total $k$-coalition: bounds, exact values and an application to double coalition

Let $G=\big{(}V(G),E(G)\big{)}$ be a graph with minimum degree $k$. A subset $S\subseteq V(G)$ is called a total $k$-dominating set if every vertex in $G$ has at least $k$ neighbors in $S$. Two disjoint sets $A,B\subset V(G)$ form a total $k$-coalition in $G$ if none of them is a total $k$-dominating set in $G$ but their union $A\cup B$ is a total $k$-dominating set. A vertex partition $Ω=\{V_{1},\ldots,V_{|Ω|}\}$ of $G$ is a total $k$-coalition partition if each set $V_{i}$ forms a total $k$-coalition with another set $V_{j}$. The total $k$-coalition number ${\rm TC}_{k}(G)$ of $G$ equals the maximum cardinality of a total $k$-coalition partition of $G$. In this paper, the above-mentioned concept are investigated from combinatorial points of view. Several sharp lower and upper bounds on ${\rm TC}_{k}(G)$ are proved, where the main emphasis is given on the invariant when $k=2$. As a consequence, the exact values of ${\rm TC}_2(G)$ when $G$ is a cubic graph or a $4$-regular graph are obtained. By using similar methods, an open question posed by Henning and Mojdeh regarding double coalition is answered. Moreover, ${\rm TC}_3(G)$ is determined when $G$ is a cubic graph.

math.CO↗

arXiv2025

On Sierpiński packing chromatic number and recognition of Sierpiński products

The Sierpiński product $G \otimes _f H$ of graphs $G$ and $H$ with respect to a function $f \colon V(G)\rightarrow V(H)$ has the vertex set $V(G)\times V(H)$. For every $g\in V(G)$ it contains a disjoint copy $gH$ of $H$, and for every edge $gg'$ of $G$ there is the edge $(g,f(g'))(g',f(g))$ between $gH$ and $g'H$. In this paper, the Sierpiński packing chromatic number is defined as the minimum of $χ_ρ(G\otimes _f H)$ over all functions $f$, where $χ_ρ(X)$ is the packing chromatic number of $X$. The upper Sierpiński packing chromatic number is analogously defined as the maximum corresponding value. The (upper) Sierpiński packing chromatic number is determined for all Sierpiński product graphs whose both factors are complete. Sierpiński product graphs whose factors are paths or stars are also studied. Their Sierpiński packing chromatic number is always $3$, while their upper Sierpiński packing chromatic number is bounded from below and above. It is also proved that for a given graph $G$, it can be checked in polynomial time whether $G$ has a representation as a Sierpiński product graphs both factors of which being trees.

math.CO↗

arXiv2025

Breaking Symmetry in Graphs by Resolving Sets

Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars attain the bound, and among connected unicyclic graphs such graphs are $t$-cycles for $t\in \{3,4,5\}$. It is shown that for any $1\leq n< m$, there exists a graph $G$ with $D(G)=n$ and ${\rm dim}(G)=m$. Using the bound $D(G) \le {\rm dim}(G)+1$, graphs with $D(G) = n(G)-2$ are classified.

math.CO↗

arXiv2025

On the local metric dimension of $K_4$-free graphs

Let $G$ be a graph of order $ n(G) $, local metric dimension $ \dim_l(G) $, and clique number $ ω(G) $. It has been conjectured that if $ n(G) \geq ω(G) + 1 \geq 4 $, then $ \dim_l(G) \leq \left( \frac{ω(G) - 2}{ω(G) - 1} \right) n(G) $. In this paper the conjecture is confirmed for the case $ ω(G) = 3 $. Consequently, a problem regarding the local metric dimension of planar graphs is also resolved.

math.CO↗

arXiv2025

S-packing chromatic critical graphs

For a non-decreasing sequence of positive integers $S=(s_1,s_2,\ldots)$, the $S$-packing chromatic number of a graph $G$ is denoted by $χ_S(G)$. In this paper, $χ_S$-critical graphs are introduced as the graphs $G$ such that $χ_S(H) < χ_S(G)$ for each proper subgraph $H$ of $G$. Several families of $χ_S$-critical graphs are constructed, and $2$- and $3$-colorable $χ_S$-critical graphs are presented for all packing sequences $S$, while $4$-colorable $χ_S$-critical graphs are found for most of $S$. Cycles which are $χ_S$-critical are characterized under different conditions. It is proved that for any graph $G$ and any edge $e \in E(G)$, the inequality $χ_S(G - e) \ge χ_S(G)/2$ holds. Moreover, in several important cases, this bound can be improved to $χ_S(G - e) \ge (χ_S(G)+1)/2$. The sharpness of the bounds is also discussed. Along the way an earlier result on $χ_S$-vertex-critical graphs is supplemented.

math.CO↗

arXiv2025

Packing chromatic number of unitary Cayley graphs of $\Bbb Z_n$ and algorithmic approaches to it

A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into $k$ disjoint non-empty classes $V_1, \dots, V_k$, such that if $u,v \in V_i$, $i\in [k]$, $u\ne v$, then the distance between $u$ and $v$ is greater than $i$. The packing chromatic number of $G$ is the smallest integer $k$ which admits a packing $k$-coloring of $G$. In this paper, the packing chromatic number of the unitary Cayley graph of $\mathbb{Z}_n$ is computed. Two metaheuristic algorithms for calculating the packing chromatic number are also proposed.

math.CO↗

arXiv2025

Varieties of mutual-visibility and general position on Sierpiński graphs

The variety of mutual-visibility problems contains four members, as does the variety of general position problems. The basic problem is to determine the cardinality of the largest such sets. In this paper, these eight invariants are investigated on Sierpiński graphs $S_p^n$. They are determined for the Sierpiński graphs $S_p^2$, $p\ge 3$. All, but the outer mutual-visibility number and the outer general position number, are also determined for $S_3^n$, $n\ge 3$. In many of the cases the corresponding extremal sets are enumerated.

math.CO↗

arXiv2025

Burning some myths on privacy properties of social networks against active attacks

This work focuses on showing some arguments addressed to dismantle the extended idea about that social networks completely lacks of privacy properties. We consider the so-called active attacks to the privacy of social networks and the counterpart $(k,\ell)$-anonymity measure, which is used to quantify the privacy satisfied by a social network against active attacks. To this end, we make use of the graph theoretical concept of $k$-metric antidimensional graphs for which the case $k=1$ represents those graphs achieving the worst scenario in privacy whilst considering the $(k,\ell)$-anonymity measure. As a product of our investigation, we present a large number of computational results stating that social networks might not be as insecure as one often thinks. In particular, we develop a large number of experiments on random graphs which show that the number of $1$-metric antidimensional graphs is indeed ridiculously small with respect to the total number of graphs that can be considered. Moreover, we search on several real networks in order to check if they are $1$-metric antidimensional, and obtain that none of them are such. Along the way, we show some theoretical studies on the mathematical properties of the $k$-metric antidimensional graphs for any suitable $k\ge 1$. In addition, we also describe some operations on graphs that are $1$-metric antidimensional so that they get embedded into another larger graphs that are not such, in order to obscure their privacy properties against active attacks.

cs.SI↗

arXiv2025

Three algorithmic approaches to the general position problem

If $G$ is a graph, then $X\subseteq V(G)$ is a general position set if for every two vertices $v,u\in X$ and every shortest $(u,v)$-path $P$, it holds that no inner vertex of $P$ lies in $X$. In this note we propose three algorithms to compute a largest general position set in $G$: an integer linear programming algorithm, a genetic algorithm, and a simulated annealing algorithm. These approaches are supported by examples from different areas of graph theory.

math.CO↗

arXiv2025

On the $Δ$-edge stability number of graphs

The $Δ$-edge stability number ${\rm es}_Δ(G)$ of a graph $G$ is the minimum number of edges of $G$ whose removal results in a subgraph $H$ with $Δ(H) = Δ(G)-1$. Sets whose removal results in a subgraph with smaller maximum degree are called mitigating sets. It is proved that there always exists a mitigating set which induces a disjoint union of paths of order $2$ or $3$. Minimum mitigating sets which induce matchings are characterized. It is proved that to obtain an upper bound of the form ${\rm es}_Δ(G) \leq c |V(G)|$ for an arbitrary graph $G$ of given maximum degree $Δ$, where $c$ is a given constant, it suffices to prove the bound for $Δ$-regular graphs. Sharp upper bounds of this form are derived for regular graphs. It is proved that if $Δ(G) \geq\frac{|V(G)|-2}{3}$ or the induced subgraph on maximum degree vertices has a $Δ(G)$-edge coloring, then ${\rm es}_Δ(G) \le \lceil |V(G)|/2\rceil$.

math.CO↗

arXiv2025

Criticality for Maker-Breaker domination games with predomination

A predominated graph is a pair $(G,D)$, where $G$ is a graph and the vertices in $D\subseteq V(G)$ are considered already dominated. Maker-Breaker domination game critical (MBD critical) predominated graphs are introduced as the predominated graphs $(G,D)$ on which Staller wins the game, but Dominator wins on $(G, D \cup \{v\})$ for every vertex $v \in V(G) \setminus D$. Tools are developed for handling the Maker-Breaker domination game on trees which lead to a characterization of Staller-win predominated trees. MBD critical predominated trees are characterized and an algorithm is designed which verifies in linear time whether a given predominated tree is MBD critical. A large class of MBD critical predominated cacti is presented and Maker-Breaker critical hypergraphs constructed.

math.CO↗

arXiv2025

Maker-Breaker domination game critical graphs

The Maker-Breaker domination game (MBD game) is a two-player game played on a graph $G$ by Dominator and Staller. They alternately select unplayed vertices of $G$. The goal of Dominator is to form a dominating set with the set of vertices selected by him while that of Staller is to prevent this from happening. In this paper MBD game critical graphs are studied. Their existence is established and critical graphs are characterized for most of the cases in which the first player can win the game in one or two moves.

math.CO↗

arXiv2025

The general position number under vertex and edge removal

Let ${\rm gp}(G)$ be the general position number of a graph $G$. It is proved that ${\rm gp}(G-x)\leq 2{\rm gp}(G)$ holds for any vertex $x$ of a connected graph $G$ and that if $x$ lies in some ${\rm gp}$-set of $G$, then ${\rm gp}(G) - 1 \le {\rm gp}(G-x)$. Constructions are given which show that ${\rm gp}(G-x)$ can be much larger than ${\rm gp}(G)$ also when $G-x$ is connected. For diameter $2$ graphs it is proved that ${\rm gp}(G-x) \le {\rm gp}(G)$, and that ${\rm gp}(G-x) \ge {\rm gp}(G) - 1$ when the diameter of $G-x$ remains $2$. It is demonstrated that ${\rm gp}(G)/2\le {\rm gp}(G-e)\leq 2{\rm gp}(G)$ holds for any edge $e$ of a graph $G$. For diameter $2$ graphs these results can be improved to ${\rm gp}(G)-1\le {\rm gp}(G-e)\leq\ {\rm gp}(G) + 1$. All these bounds are proved to be sharp.

math.CO↗