arXiv subjects

Merce Mora

Publications and source records attributed to Merce Mora.

6 recordsLinked to original sources

arXiv2026

Isolation subdivision number of a graph

For a graph $G=(V,E),$ a set $S \subseteq V$ is called an isolating set of $G$ if the set $V-N[S]$ is independent. The minimum cardinality of an isolating set in $G$ is the isolation number of $G$, denoted by $\iota(G).$ Here we introduce the isolation subdivision number of a graph $G$, denoted by ${\rm sd}_\iota(G)$, as the minimum number of edges of $G$ that must be subdivided, where each edge can be subdivided at most once, in order to obtain a graph with isolation number greater than $\iota(G).$ We show that the new parameter is well defined for any non-trivial graph different from a star and that it can be arbitrarily large. We present the values of this parameter for some elementary classes of graphs and establish some basic properties. We show also that $1\leq {\rm sd}_\iota(T)\leq 4$ for any tree $T$ different from a star and characterize all trees $T$ with ${\rm sd}_\iota(T)=1.$

math.CO↗

arXiv2018

Neighbor-Locating Colorings in Graphs

A $k$-coloring of a graph $G$ is a $k$-partition $\Pi=\{S_1,\ldots,S_k\}$ of $V(G)$ into independent sets, called \emph{colors}. A $k$-coloring is called \emph{neighbor-locating} if for every pair of vertices $u,v$ belonging to the same color $S_i$, the set of colors of the neighborhood of $u$ is different from the set of colors of the neighborhood of $v$. The neighbor-locating chromatic number $\chi _{_{NL}}(G)$ is the minimum cardinality of a neighbor-locating coloring of $G$. We establish some tight bounds for the neighbor-locating chromatic number of a graph, in terms of its order, maximum degree and independence number. We determine all connected graphs of order $n\geq 5$ with neighbor-locating chromatic number $n$ or $n-1$. We examine the neighbor-locating chromatic number for two graph operations: join and disjoint union, and also for two graph families: split graphs and Mycielski graphs.

math.CO↗

arXiv2016

On the Partition Dimension and the Twin Number of a Graph

A partition P of the vertex set of a connected graph G is a locating partition of G if every vertex is uniquely determined by its vector of distances to the elements of P. The partition dimension of G is the minimum cardinality of a locating partition of G. A pair of vertices u,v of a graph G are called twins if they have exactly the same set of neighbors other than u and v. A twin class is a maximal set of pairwise twin vertices. The twin number of a graph G is the maximum cardinality of a twin class of G. In this paper we undertake the study of the partition dimension of a graph by also considering its twin number. This approach allows us to obtain the set of connected graphs of order n having partition dimension n-2. This set is formed by exactly 15 graphs, instead of 23, as was wrongly stated in the paper: "Discrepancies between metric dimension and partition dimension of a connected graph", published in Discrete Mathematics in 2008.

math.CO↗

arXiv2015

On global location-domination in bipartite graphs

A dominating set $S$ of a graph $G$ is called locating-dominating, LD-set for short, if every vertex $v$ not in $S$ is uniquely determined by the set of neighbors of $v$ belonging to $S$. Locating-dominating sets of minimum cardinality are called $LD$-codes and the cardinality of an LD-code is the \emph{location-domination number} $\lambda(G)$. An LD-set $S$ of a graph $G$ is \emph{global} if it is an LD-set of both $G$ and its complement $\overline{G}$. The \emph{global location-domination number} $\lambda_g(G)$ is the minimum cardinality of a global LD-set of $G$. For any LD-set $S$ of a given graph $G$, the so-called \emph{S-associated graph} $G^S$ is introduced. This edge-labeled bipartite graph turns out to be very helpful to approach the study of LD-sets in graphs, particularly when $G$ is bipartite. This paper is mainly devoted to the study of relationships between global LD-sets, LD-codes and the location-domination number in a graph $G$ and its complement $\overline{G}$, when $G$ is bipartite.

math.CO↗

arXiv2009

On the geodetic and the hull numbers in strong product graphs

A set S of vertices of a connected graph G is convex, if for any pair of vertices u; v 2 S, every shortest path joining u and v is contained in S . The convex hull CH(S) of a set of vertices S is defined as the smallest convex set in G containing S. The set S is geodetic, if every vertex of G lies on some shortest path joining two vertices in S, and it is said to be a hull set if its convex hull is V(G). The geodetic and the hull numbers of G are the cardinality of a minimum geodetic and a minimum hull set, respectively. In this work, we investigate the behavior of both geodetic and hull sets with respect to the strong product operation for graphs. We also stablish some bounds for the geodetic number and the hull number and obtain the exact value of these parameters for a number of strong product graphs.

math.CO↗

arXiv2007

Extremal Graph Theory for Metric Dimension and Diameter

A set of vertices $S$ \emph{resolves} a connected graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The \emph{metric dimension} of $G$ is the minimum cardinality of a resolving set of $G$. Let $\mathcal{G}_{\beta,D}$ be the set of graphs with metric dimension $\beta$ and diameter $D$. It is well-known that the minimum order of a graph in $\mathcal{G}_{\beta,D}$ is exactly $\beta+D$. The first contribution of this paper is to characterise the graphs in $\mathcal{G}_{\beta,D}$ with order $\beta+D$ for all values of $\beta$ and $D$. Such a characterisation was previously only known for $D\leq2$ or $\beta\leq1$. The second contribution is to determine the maximum order of a graph in $\mathcal{G}_{\beta,D}$ for all values of $D$ and $\beta$. Only a weak upper bound was previously known.

math.CO↗