arXiv subjects

Mercè Mora

Publications and source records attributed to Mercè Mora.

At least 19 recordsLinked to original sources

arXiv2026

On Generalized Token Graphs

The vertices of a $k$-token graph of a graph $G$ correspond to $k$ indistinguishable tokens placed on $k$ different vertices of $G$. Changing some conditions on both the nature of the tokens and the number of tokens allowed in each vertex of $G$, we define a generalization of token graphs, which we call generalized token graphs or simply supertoken graphs, which have different applications. Depending on the above conditions, different families of graphs (such as the Cartesian $k$-th power of $G$ by itself) are obtained, and we present some of their properties, including order, size, and connectivity.

math.CO↗

arXiv2026

Metric representations by minimal graphs

A resolving set in a graph $G$ is a vertex subset $W= \{ω^1, \dots, ω^n\} \subseteq V(G)$ such that each $u \in V(G)$ can be uniquely identified by the vector $r(u \vert W) = (d(u,ω^1), \dots, d(u,ω^n))$ of metric coordinates of $u$ with respect to $W$. The reverse problem of identifying the vector sets that are a set of coordinates of some graph provides the concept of realizable vector set $S \subset \mathbb{Z}^n$ by a pair $(G, W)$ meaning that $S=\{ r(u\vert W)\colon u\in V(G)\}$ with $W$ a resolving set of the graph $G$. Here we focus on the minimality of the realizations of vector sets with respect to their edge sets. On the one hand, we study conditions under which it is possible to remove an edge from the graph and keep the realizability condition. This provides a method for finding minimal realizations, as well as allowing us to characterize uniquely realizable vector sets. On the other hand, we prove that the decision problem of realizing a vector set by a graph with a given number of edges is an NP-complete problem. Finally, we characterize the vector sets that are realizable by a tree and, furthermore, we study the case in which such a realization is the only one.

math.CO↗

arXiv2025

Upper bounds on the $k$-isolation number

The isolation number of a graph $G$ (also called the vertex-edge domination number of $G$), denoted by $ι(G)$, is the size of a smallest subset $D$ of the vertex set $V(G)$ of $G$ such that $G-N[D]$ (the graph obtained by deleting the closed neighbourhood $N[D]$ of $D$ from $G$) has no edges. For $k \geq 1$, the $k$-isolation number of $G$ is the size of a smallest subset $D$ of $V(G)$ such that the maximum degree of $G-N[D]$ is at most $k-1$. Thus, $ι_1(G) = ι(G)$. Let $n$ and $\ell$ be the number of vertices and the number of leaves of $G$, respectively. We show that if $n \geq 3$ and $G$ is connected, then $ι_k(G) \leq \frac{n - \ell}{2}$. We also show that if $G$ is a tree $T$, then $ι(T) \leq \frac{n + \ell}{4}$ and $ι_k(T) \leq \frac{n + \ell}{2k+1}$ for $k \geq 2$. These bounds together improve the inequality $ι_k(T) \leq \frac{n}{k+2}$ of Caro and Hansberg except that their inequality is better if $k \geq 2$ and $\frac{k-1}{k+2}n < \ell < \frac{k}{k+2}n$. Each of the new bounds is attainable if it is an integer. For each of them, we characterize all the graphs that attain it.

math.CO↗

arXiv2024

On the metric representation of the vertices of a graph

The metric representation of a vertex $u$ in a connected graph $G$ respect to an ordered vertex subset $W=\{ω_1, \dots , ω_n\}\subset V(G)$ is the vector of distances $r(u\vert W)=(d(u,ω_1), \dots , d(u,ω_n))$. A vertex subset $W$ is a resolving set of $G$ if $r(u\vert W)\neq r(v\vert W)$, for every $u,v\in V(G)$ with $u\neq v$. Thus, a resolving set with $n$ elements provides a set of metric representation vectors $S\subset \mathbb{Z}^n$ with cardinal equal to the order of the graph. In this paper, we address the reverse point of view, that is, we characterize the finite subsets $S\subset \mathbb{Z}^n$ that are realizable as the set of metric representation vectors of a graph $G$ with respect to some resolving set $W$. We also explore the role that the strong product of paths plays in this context. Moreover, in the case $n=2$, we characterize the sets $S\subset \mathbb{Z}^2$ that are uniquely realizable as the set of metric representation vectors of a graph $G$ with respect to a resolving set $W$.

math.CO↗

arXiv2024

Antimagic and product antimagic graphs with pendant edges

Let $G=(V,E)$ be a simple graph of size $m$ and $L$ a set of $m$ distinct real numbers. An $L$-labeling of $G$ is a bijection $ϕ: E \rightarrow L$. We say that $ϕ$ is an antimagic $L$-labeling if the induced vertex sum $ϕ_+: V \rightarrow \mathbb {R}$ defined as $ϕ_+(u)=\sum_{uv\in E}ϕ(uv)$ is injective. Similarly, $ϕ$ is a product antimagic $L$-labeling of $G$ if the induced vertex product $ϕ_{\circ}: V \rightarrow \mathbb {R}$ defined as $ϕ_{\circ}(u)=\prod_{uv\in E}ϕ(uv)$ is injective. A graph $G$ is antimagic (resp. product antimagic) if it has an antimagic (resp. a product antimagic) $L$-labeling for $L=\{1,2,\dots,m\}$. Hartsfield and Ringel conjectured that every simple connected graph distinct from $K_2$ is antimagic, but the conjecture remains widely open. We prove, among other results, that every connected graph of size $m$, $m \geq 3$, admits an antimagic $L$-labeling for every arithmetic sequence $L$ of $m$ positive real numbers, if every vertex of degree at least three is a support vertex. As a corollary, we derive that these graphs are antimagic, reinforcing the veracity of the conjecture by Hartsfield and Ringel. Moreover, these graphs admit also a product antimagic $L$-labeling provided that the smallest element of $L$ is at least one. The proof is constructive.

math.CO↗

arXiv2023

Graphs with isolation number equal to one third of the order

A set $D$ of vertices of a graph $G$ is isolating if the set of vertices not in $D$ or with no neighbor in $D$ is independent. The isolation number of $G$, denoted by $ι(G)$, is the minimum cardinality of an isolating set of $G$. It is known that $ι(G)\le n/3$, if $G$ is a connected graph of order $n$, $n\ge 3$, distinct from $C_5$. The main result of this work is the characterisation of unicyclic and block graphs of order $n$ with isolating number equal to $n/3$. Moreover, we provide a family of general graphs attaining this upper bound on the isolation number.

math.CO↗

arXiv2021

Resolving sets tolerant to failures in three-dimensional grids

An ordered set $S$ of vertices of a graph $G$ is a resolving set for $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of G is the minimum cardinality of a resolving set. In this paper we study resolving sets tolerant to several failures in three-dimensional grids. Concretely, we seek for minimum cardinality sets that are resolving after removing any $k$ vertices from the set. This is equivalent to finding $(k+1)$-resolving sets, a generalization of resolving sets, where, for every pair of vertices, the vector of distances to the vertices of the set differ in at least $k+1$ coordinates. This problem is also related with the study of the $(k+1)$-metric dimension of a graph, defined as the minimum cardinality of a $(k+1)$-resolving set. In this work, we first prove that the metric dimension of a three-dimensional grid is 3 and establish some properties involving resolving sets in these graphs. Secondly, we determine the values of $k\ge 1$ for which there exists a $(k+1)$-resolving set and construct such a resolving set of minimum cardinality in almost all cases.

math.CO↗

arXiv2020

Reappraising the distribution of the number of edge crossings of graphs on a sphere

Many real transportation and mobility networks have their vertices placed on the surface of the Earth. In such embeddings, the edges laid on that surface may cross. In his pioneering research, Moon analyzed the distribution of the number of crossings on complete graphs and complete bipartite graphs whose vertices are located uniformly at random on the surface of a sphere assuming that vertex placements are independent from each other. Here we revise his derivation of that variance in the light of recent theoretical developments on the variance of crossings and computer simulations. We show that Moon's formulae are inaccurate in predicting the true variance and provide exact formulae.

cs.DM↗

arXiv2020

The Neighbor-Locating-Chromatic Number of Pseudotrees

A $k$-coloring of a graph $G$ is a partition of the set of vertices of $G$ into $k$ independent sets, which are called colors. A $k$-coloring is neighbor-locating if any two vertices belonging to the same color can be distinguished from each other by the colors of their respective neighbors. The neighbor-locating chromatic number $χ_{_{NL}}(G)$ is the minimum cardinality of a neighbor-locating coloring of $G$. In this paper, we determine the neighbor-locating chromatic number of paths, cycles, fans, and wheels. Moreover, a procedure to construct a neighbor-locating coloring of minimum cardinality for these families of graphs is given. We also obtain tight upper bounds on the order of trees and unicyclic graphs in terms of the neighbor-locating chromatic number. Further partial results for trees are also established.

math.CO↗

arXiv2019

Trees whose even-degree vertices induce a path are antimagic

An antimagic labeling a connected graph $G$ is a bijection from the set of edges $E(G)$ to $\{1,2,\dots,|E(G)|\}$ such that all vertex sums are pairwise distinct, where the vertex sum at vertex $v$ is the sum of the labels assigned to edges incident to $v$. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that every simple connected graph other than $K_2$ is antimagic; however, the conjecture remains open, even for trees. In this note we prove that trees whose vertices of even degree induce a path are antimagic, extending a result given by Liang, Wong, and Zhu [Discrete Math. 331 (2014) 9--14].

math.CO↗

arXiv2019

Metric dimension of maximal outerplanar graphs

In this paper, we study the metric dimension problem in maximal outerplanar graphs. Concretely, if $β(G)$ is the metric dimension of a maximal outerplanar graph $G$ of order $n$, we prove that $2\le β(G) \le \lceil \frac{2n}{5}\rceil$ and that the bounds are tight. We also provide linear algorithms to decide whether the metric dimension of $G$ is 2 and to build a resolving set of size $\lceil \frac{2n}{5}\rceil$ for $G$. Moreover, we characterize the maximal outerplanar graphs with metric dimension 2.

math.CO↗

arXiv2019

Caterpillars are Antimagic

An antimagic labeling of a graph $G$ is an injection from $E(G)$ to $\{1,2,\dots,|E(G)|\}$ such that all vertex sums are pairwise distinct, where the vertex sum at vertex $u$ is the sum of the labels assigned to edges incident to $u$. A graph is called antimagic when it has an antimagic labeling. Hartsfield and Ringel conjectured that every simple connected graph other than $K_2$ is antimagic and the conjecture remains open even for trees. Here we prove that caterpillars are antimagic by means of an $O(n \log n)$ algorithm.

math.CO↗

arXiv2019

Antimagic Labelings of Caterpillars

A $k$-antimagic labeling of a graph $G$ is an injection from $E(G)$ to $\{1,2,\dots,|E(G)|+k\}$ such that all vertex sums are pairwise distinct, where the vertex sum at vertex $u$ is the sum of the labels assigned to edges incident to $u$. We call a graph $k$-antimagic when it has a $k$-antimagic labeling, and antimagic when it is 0-antimagic. Hartsfield and Ringel conjectured that every simple connected graph other than $K_2$ is antimagic, but the conjecture is still open even for trees. Here we study $k$-antimagic labelings of caterpillars, which are defined as trees the removal of whose leaves produces a path, called its spine. As a general result, we use constructive techniques to prove that any caterpillar of order $n$ is $(\lfloor (n-1)/2 \rfloor - 2)$-antimagic. Furthermore, if $C$ is a caterpillar with a spine of order $s$, we prove that when $C$ has at least $\lfloor (3s+1)/2 \rfloor$ leaves or $\lfloor (s-1)/2 \rfloor$ consecutive vertices of degree at most 2 at one end of a longest path, then $C$ is antimagic. As a consequence of a result by Wong and Zhu, we also prove that if $p$ is a prime number, any caterpillar with a spine of order $p$, $p-1$ or $p-2$ is $1$-antimagic.

math.CO↗

arXiv2018

Resolving dominating partitions in graphs

A partition $Π=\{S_1,\ldots,S_k\}$ of the vertex set of a connected graph $G$ is called a \emph{resolving partition} of $G$ if for every pair of vertices $u$ and $v$, $d(u,S_j)\neq d(v,S_j)$, for some part $S_j$. The \emph{partition dimension} $β_p(G)$ is the minimum cardinality of a resolving partition of $G$. A resolving partition $Π$ is called \emph{resolving dominating} if for every vertex $v$ of $G$, $d(v,S_j)=1$, for some part $S_j$ of $Π$. The \emph{dominating partition dimension} $η_p(G)$ is the minimum cardinality of a resolving dominating partition of $G$. In this paper we show, among other results, that $β_p(G) \le η_p(G) \le β_p(G)+1$. We also characterize all connected graphs of order $n\ge7$ satisfying any of the following conditions: $η_p(G)= n$, $η_p(G)= n-1$, $η_p(G)= n-2$ and $β_p(G) = n-2$. Finally, we present some tight Nordhaus-Gaddum bounds for both the partition dimension $β_p(G)$ and the dominating partition dimension $η_p(G)$.

math.CO↗

arXiv2018

General bounds on limited broadcast domination

Dominating broadcasting is a domination-type structure that models a transmission antenna network. In this paper, we study a limited version of this structure, that was proposed as a common framework for both broadcast and classical domination. In this limited version, the broadcast function is upper bounded by an integer $k$ and the minimum cost of such function is the dominating $k$-broadcast number. Our main result is a unified upper bound on this parameter for any value of $k$ in general graphs, in terms of both $k$ and the order of the graph. We also study the computational complexity of the associated decision problem.

math.CO↗

arXiv2018

Locating domination in bipartite graphs and their complements

A set $S$ of vertices of a graph $G$ is \emph{distinguishing} if the sets of neighbors in $S$ for every pair of vertices not in $S$ are distinct. A \emph{locating-dominating set} of $G$ is a dominating distinguishing set. The \emph{location-domination number} of $G$, $λ(G)$, is the minimum cardinality of a locating-dominating set. In this work we study relationships between $λ({G})$ and $λ(\overline{G})$ for bipartite graphs. The main result is the characterization of all connected bipartite graphs $G$ satisfying $λ(\overline{G})=λ({G})+1$. To this aim, we define an edge-labeled graph $G^S$ associated with a distinguishing set $S$ that turns out to be very helpful.

math.CO↗

arXiv2017

Dominating 2-broadcast in graphs: complexity, bounds and extremal graphs

Limited dominating broadcasts were proposed as a variant of dominating broadcasts, where the broadcast function is upper bounded. As a natural extension of domination, we consider dominating $2$-broadcasts along with the associated parameter, the dominating $2$-broadcast number. We prove that computing the dominating $2$-broadcast number is a NP-complete problem, but can be achieved in linear time for trees. We also give an upper bound for this parameter, that is tight for graphs as large as desired.

math.CO↗

arXiv2016

Uniform hypergraphs and dominating sets of graphs

A (simple) hypergraph is a family H of pairwise incomparable sets of a finite set. We say that a hypergraph H is a domination hypergraph if there is at least a graph G such that the collection of minimal dominating sets of G is equal to H. Given a hypergraph, we are interested in determining if it is a domination hypergraph and, if this is not the case, we want to find domination hypergraphs in some sense close to it, the domination completions. Here we will focus on the family of hypergraphs containing all the subsets with the same cardinality, the uniform hypergraphs of maximum size. Specifically, we characterize those hypergraphs H in this family that are domination hypergraphs and, in any other case, we prove that the hypergraph H is uniquely determined by some of its domination completions and that H can be recovered from them by using a suitable hypergraph operation.

math.CO↗