Search arXiv⌕ Search

arXiv subjects

Dorota Kuziak

Publications and source records attributed to Dorota Kuziak.

At least 19 recordsLinked to original sources
arXiv2026

A variety of the mutual-visibility coloring problem for graphs

This paper explores variations of vertex-coloring problems defined on graph visibility properties. It introduces and studies the dual, outer, and total mutual-visibility chromatic numbers, which partition the vertex set of a graph into color classes that preserve specific mutual-visibility conditions called dual, outer or total. The work provides structural conditions under which these chromatic parameters are finite or infinite, and establishes that deciding whether a graph admits a dual, outer, or total mutual-visibility coloring using a given number of colors is NP-complete, even when restricted to two colors. Exact formulas and tight bounds for these chromatic parameters are established across several fundamental graph classes. For block graphs, complete characterizations are provided for the outer and dual mutual-visibility chromatic numbers based on structural invariants such as cut vertices and specific forbidden subgraph structures. On Hamming graphs, the dual and total mutual-visibility chromatic numbers are shown to equal the smaller dimension of the factors, while the outer mutual-visibility chromatic number is proven to equal the star arboricity of a corresponding complete bipartite graph. Finally, the paper examines strong grid graphs, determining exact values for their total, outer, and dual mutual-visibility chromatic numbers. These results demonstrate how the parameter behaviors range from finite constants to infinity depending on the grid dimensions.

math.CO↗
arXiv2026

Vertices that belong to every minimum dominating set of a graph and their connection with transportation sharing systems with study cases in Campo de Gibraltar area

This study addresses a theoretical model regarding equity and accessibility challenges in designing shared transportation systems (such as micro-mobility networks) by applying graph-vertex domination setting. The work focuses on identifying dominating forced vertices, that represent nodes belonging to every dominating set of a graph of the smallest possible cardinality, and which correspond to critical, non-negotiable station locations essential for maintaining system efficiency and coverage. From a theoretical perspective, in the paper it is first demonstrated that determining whether a given vertex is a dominating forced vertex is co-NP-hard, establishing the computational infeasibility of exact identification in large networks. To analyze graph structures, sharp theoretical bounds on the maximum number of dominating forced vertices are established, proving that their count is bounded above by one-third of the order of the graph, and provide complete structural characterizations for graphs achieving this bound, as well as, trees with no dominating forced vertices. To overcome computational limits in practical urban settings, the study uses an iterated greedy metaheuristic framework to generate minimal dominating sets and approximate critical forced nodes based on their appearance frequency across iterations. The methodology is validated on strong grid graphs and applied to real-world road network models of Algeciras and La Línea de la Concepción, two cities in the area of Campo de Gibraltar, Spain, which successfully pinpoints candidate location points for scooter-sharing stations across both cities.

physics.soc-ph↗
arXiv2026

Multiset resolvability parameters in graphs: A survey with new results and open problems

The metric dimension, which has lots of variants and numerous applications in other fields, is one of the most important and most extensively studied topics in metric graph theory. Results in which resolvability is achieved by considering multisets of distances from a fixed vertex, instead of vectors as in the original version, are surveyed. The concepts discussed are multiset dimension, outer multiset dimension, local multiset dimension, edge multiset dimension, and $k$-multiset antidimension. Along the way, sharp lower bounds on the outer multiset dimension of diameter two graphs and join graphs with edgeless graphs are proved, which solves two open problems from the literature. New results on graphs with local multiset dimension equal to two are also proved. In particular, such graphs are characterized among block graphs. Finally, a list of open problems from the literature is compiled, and several new problems are added to the list for future research.

math.CO↗
arXiv2026

The weak $k$-metric dimension of the direct product of complete graphs

The weak $k$-metric dimension of a graph is roughly understood as the cardinality of a smallest set of vertices $S$ of the graph with the property of uniquely recognizing all the vertices of the graph throughout summations of differences of distances to the vertices of $S$. The weak $k$-metric dimension of the direct product of two isomorphic complete graphs is considered in this work. Specifically, the value of such parameter is computed for almost all possibilities of these products and a bound is provided in the remaining case.

math.CO↗
arXiv2025

Moving through Cartesian products, coronas and joins in general position

The general position problem asks for large sets of vertices such that no three vertices of the set lie on a common shortest path. Recently a dynamic version of this problem was defined, called the \emph{mobile general position problem}, in which a collection of robots must visit all the vertices of the graph whilst remaining in general position. In this paper we investigate this problem in the context of Cartesian products, corona products and joins, giving upper and lower bounds for general graphs and exact values for families including grids, cylinders, Hamming graphs and prisms of trees.

math.CO↗
arXiv2025

On the $(k,\ell)$-multiset anonymity measure for social graphs

The publication of social graphs must be preceded by a rigorous analysis of privacy threats against social graph users. When the threat comes from inside the social network itself, the threat is called an active attack, and the de-facto privacy measure used to quantify the resistance to such an attack is the $(k,\ell)$-anonymity. The original formulation of $(k,\ell)$-anonymity represents the adversary's knowledge as a vector of distances to the set of attacker nodes. In this article, we argue that such adversary is too strong when it comes to counteracting active attacks. We, instead, propose a new formulation where the adversary's knowledge is the multiset of distances to the set of attacker nodes. The goal of this article is to study the $(k,\ell)$-multiset anonymity from a graph theoretical point of view, while establishing its relationship to $(k,\ell)$-anonymity in one hand, and considering the $k$-multiset antiresolving sets as its theoretical frame, in a second one. That is, we prove properties of some graph families in relation to whether they contain a set of attacker nodes that breaks the $(k,\ell)$-multiset anonymity. From a practical point of view, we develop a linear programming formulation of the $k$-multiset antiresolving sets that allows us to calculate the resistance of social graphs against active attacks. This is useful for analysts who wish to know the level of privacy offered by a graph.

math.CO↗
arXiv2025

On the weak $k$-metric dimension of Hamming graphs

Given a connected graph $G$, a set of vertices $X\subset V(G)$ is a weak $k$-resolving set of $G$ if for each two vertices $y,z\in V(G)$, the sum of the values $|d_G(y,x)-d_G(z,x)|$ over all $x\in X$ is at least $k$, where $d_G(u,v)$ stands for the length of a shortest path between $u$ and $v$. The cardinality of a smallest weak $k$-resolving set of $G$ is the weak $k$-metric dimension of $G$, and is denoted by $\mathrm{wdim}_k(G)$. In this paper, $\mathrm{wdim}_k(K_n\,\square\,K_n)$ is determined for every $n\ge 3$ and every $2\le k\le 2n$. An improvement of a known integer linear programming formulation for this problem is developed and implemented for the graphs $K_n\,\square\,K_m$. Conjectures regarding these general situations are posed.

math.CO↗
arXiv2024

General position problems in strong and lexicographic products of graphs

Outer, dual, and total general position sets are studied on strong and lexicographic products of graphs. Sharp lower and upper bounds are proved for the outer and the dual general position number of strong products and several exact values are obtained. For the lexicographic product, the outer general position number is determined in all the cases, and the dual general position number in many cases. The total general position number is determined for both products. Along the way some results on outer general position sets are also derived.

math.CO↗
arXiv2024

Coloring the vertices of a graph with mutual-visibility property

Given a graph $G$, a mutual-visibility coloring of $G$ is introduced as follows. We color two vertices $x,y\in V(G)$ with a same color, if there is a shortest $x,y$-path whose internal vertices have different colors than $x,y$. The smallest number of colors needed in a mutual-visibility coloring of $G$ is the mutual-visibility chromatic number of $G$, which is denoted $χ_μ(G)$. Relationships between $χ_μ(G)$ and its two parent ones, the chromatic number and the mutual-visibility number, are presented. Graphs of diameter two are considered, and in particular the asymptotic growth of the mutual-visibility number of the Cartesian product of complete graphs is determined. A greedy algorithm that finds a mutual-visibility coloring is designed and several possible scenarios on its efficiency are discussed. Several bounds are given in terms of other graph parameters such as the diameter, the order, the maximum degree, the degree of regularity of regular graphs, and/or the mutual-visibility number. For the corona products it is proved that the value of its mutual-visibility chromatic number depends on that of the first factor of the product. Graphs $G$ for which $χ_μ(G)=2$ are also considered.

math.CO↗
arXiv2024

Maker-Breaker resolving game played on corona products of graphs

The Maker-Breaker resolving game is a game played on a graph $G$ by Resolver and Spoiler. The players taking turns alternately in which each player selects a not yet played vertex of $G$. The goal of Resolver is to select all the vertices in a resolving set of $G$, while that of Spoiler is to prevent this from happening. The outcome $o(G)$ of the game played is one of $\mathcal{R}$, $\mathcal{S}$, and $\mathcal{N}$, where $o(G)=\mathcal{R}$ (resp.\ $o(G)=\mathcal{S}$), if Resolver (resp.\ Spoiler) has a winning strategy no matter who starts the game, and $o(G)=\mathcal{N}$, if the first player has a winning strategy. In this paper, the game is investigated on corona products $G\odot H$ of graphs $G$ and $H$. It is proved that if $o(H)\in\{\mathcal{N}, \mathcal{S}\}$, then $o(G\odot H) = \mathcal{S}$. No such result is possible under the assumption $o(H) = \mathcal{R}$. It is proved that $o(G\odot P_k) = \mathcal{S}$ if $k=5$, otherwise $o(G\odot P_k) = \mathcal{R}$, and that $o(G\odot C_k) = \mathcal{S}$ if $k=3$, otherwise $o(G\odot C_k) = \mathcal{R}$. Several results are also given on corona products in which the second factor is of diameter at most $2$.

math.CO↗
arXiv2023

Total mutual-visibility in graphs with emphasis on lexicographic and Cartesian products

Given a connected graph $G$, the total mutual-visibility number of $G$, denoted $μ_t(G)$, is the cardinality of a largest set $S\subseteq V(G)$ such that for every pair of vertices $x,y\in V(G)$ there is a shortest $x,y$-path whose interior vertices are not contained in $S$. Several combinatorial properties, including bounds and closed formulae, for $μ_t(G)$ are given in this article. Specifically, we give several bounds for $μ_t(G)$ in terms of the diameter, order and/or connected domination number of $G$ and show characterizations of the graphs achieving the limit values of some of these bounds. We also consider those vertices of a graph $G$ that either belong to every total mutual-visibility set of $G$ or does not belong to any of such sets, and deduce some consequences of these results. We determine the exact value of the total mutual-visibility number of lexicographic products in terms of the orders of the factors, and the total mutual-visibility number of the first factor in the product. Finally, we give some bounds and closed formulae for the total mutual-visibility number of Cartesian product graphs.

math.CO↗
arXiv2023

On the (k,l)-anonymity of networks via their $k$-metric antidimension

This work focuses on the (k,l)-anonymity of some networks as a measure of their privacy against active attacks. Two different types of networks are considered. The first one consists of graphs with a predetermined structure, namely cylinders, toruses, and $2$-dimensional Hamming graphs, whereas the second one is formed by randomly generated graphs. In order to evaluate the (k,l)-anonymity of the considered graphs, we have computed their k-metric antidimension. To this end, we have taken a combinatorial approach for the graphs with a predetermined structure, whereas for randomly generated graphs we have developed an integer programming formulation and computationally tested its implementation. The results of the combinatorial approach, as well as those from the implementations indicate that, according to the (k,l)-anonymity measure, only the 2-dimensional Hamming graphs and some general random dense graphs are achieving some higher privacy properties.

math.OC↗
arXiv2022

Nonlocal metric dimension of graphs

Nonlocal metric dimension ${\rm dim}_{\rm n\ell}(G)$ of a graph $G$ is introduced as the cardinality of a smallest nonlocal resolving set, that is, a set of vertices which resolves each pair of non-adjacent vertices of $G$. Graphs $G$ with ${\rm dim}_{\rm n\ell}(G) = 1$ or with ${\rm dim}_{\rm n\ell}(G) = n(G)-2$ are characterized. The nonlocal metric dimension is determined for block graphs, for corona products, and for wheels. Two upper bounds on the nonlocal metric dimension are proved. An embedding of an arbitrary graph into a supergraph with a small nonlocal metric dimension and small diameter is presented.

math.CO↗
arXiv2022

Further contributions on the outer multiset dimension of graphs

The outer multiset dimension ${\rm dim}_{\rm ms}(G)$ of a graph $G$ is the cardinality of a smallest set of vertices that uniquely recognize all the vertices outside this set by using multisets of distances to the set. It is proved that ${\rm dim}_{\rm ms}(G) = n(G) - 1$ if and only if $G$ is a regular graph with diameter at most $2$. Graphs $G$ with ${\rm dim}_{\rm ms}(G)=2$ are described and recognized in polynomial time. A lower bound on the lexicographic product of $G$ and $H$ is proved when $H$ is complete or edgeless, and the extremal graphs are determined. It is proved that ${\rm dim}_{\rm ms}(P_s\,\square\, P_t) = 3$ for $s\ge t\ge 2$.

math.CO↗
arXiv2021

Relating the outer-independent total Roman domination number with some classical parameters of graphs

For a given graph $G$ without isolated vertex we consider a function $f: V(G) \rightarrow \{0,1,2\}$. For every $i\in \{0,1,2\}$, let $V_i=\{v\in V(G):\; f(v)=i\}$. The function $f$ is known to be an outer-independent total Roman dominating function for the graph $G$ if it is satisfied that; (i) every vertex in $V_0$ is adjacent to at least one vertex in $V_2$; (ii) $V_0$ is an independent set; and (iii) the subgraph induced by $V_1\cup V_2$ has no isolated vertex. The minimum possible weight $ω(f)=\sum_{v\in V(G)}f(v)$ among all outer-independent total Roman dominating functions for $G$ is called the outer-independent total Roman domination number of $G$. In this article we obtain new tight bounds for this parameter that improve some well-known results. Such bounds can also be seen as relationships between this parameter and several other classical parameters in graph theory like the domination, total domination, Roman domination, independence, and vertex cover numbers. In addition, we compute the outer-independent total Roman domination number of Sierpiński graphs, circulant graphs, and the Cartesian and direct products of complete graphs.

math.CO↗
arXiv2021

A Steiner general position problem in graph theory

Let $G$ be a graph. The Steiner distance of $W\subseteq V(G)$ is the minimum size of a connected subgraph of $G$ containing $W$. Such a subgraph is necessarily a tree called a Steiner $W$-tree. The set $A\subseteq V(G)$ is a $k$-Steiner general position set if $V(T_B)\cap A = B$ holds for every set $B\subseteq A$ of cardinality $k$, and for every Steiner $B$-tree $T_B$. The $k$-Steiner general position number ${\rm sgp}_k(G)$ of $G$ is the cardinality of a largest $k$-Steiner general position set in $G$. Steiner cliques are introduced and used to bound ${\rm sgp}_k(G)$ from below. The $k$-Steiner general position number is determined for trees, cycles and joins of graphs. Lower bounds are presented for split graphs, infinite grids and lexicographic products. The lower bound for the latter products leads to an exact formula for the general position number of an arbitrary lexicographic product.

math.CO↗
arXiv2020

Dominating the direct product of two graphs through total Roman strategies

Given a graph $G$ without isolated vertices, a total Roman dominating function for $G$ is a function $f : V(G)\rightarrow \{0,1,2\}$ such that every vertex with label 0 is adjacent to a vertex with label 2, and the set of vertices with positive labels induces a graph of minimum degree at least one. The total Roman domination number $γ_{tR}(G)$ of $G$ is the smallest possible value of $\sum_{v\in V(G)}f(v)$ among all total Roman dominating functions $f$. The total Roman domination number of the direct product $G\times H$ of the graphs $G$ and $H$ is studied in this work. Specifically, several relationships, in the shape of upper and lower bounds, between $γ_{tR}(G\times H)$ and some classical domination parameters for the factors are given. Characterizations of the direct product graphs $G\times H$ achieving small values ($\le 7$) for $γ_{tR}(G\times H)$ are presented, and exact values for $γ_{tR}(G\times H)$ are deduced, while considering various specific direct product classes.

math.CO↗