Publications (50)18.9 Total impact

Article: On independent $[1,2]$sets in trees
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ABSTRACT: An independent $[1,k]$set $S$ in a graph $G$ is a dominating set which is independent and such that every vertex not in $S$ has at most $k$ neighbors in it. The existence of such sets is not guaranteed in every graph and trees having an independent $[1,k]$set have been characterized. In this paper we solve some problems previously posed by other authors about independent $[1,2]$sets. We provide a necessary condition for a graph to have an independent $[1,2]$set, in terms of spanning trees and we prove that this condition is also sufficient for cactus graphs. We follow the concept of excellent tree and characterize the family of trees such that any vertex belong to some independent $[1,2]$set. Finally we describe a linear algorithm to decide whether a tree has an independent $[1,2]$set. Such algorithm can be easily modified to obtain the cardinality of the smallest independent $[1,2]$set of a tree.  [Show abstract] [Hide abstract]
ABSTRACT: A $k$quasiperfect dominating set ($k\ge 1$) of a graph $G$ is a vertex subset $S$ such that every vertex not in $S$ is adjacent to at least one and at most k vertices in $S$. The cardinality of a minimum kquasiperfect dominating set in $G$ is denoted by $\gamma_{\stackrel{}{1k}}(G)$. Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept. The quasiperfect domination chain $\gamma_{\stackrel{}{11}}(G)\ge\gamma_{\stackrel{}{12}}(G)\ge\dots\ge\gamma_{\stackrel{}{1\Delta}}(G)=\gamma(G)$, indicates what it is lost in size when you move towards a more perfect domination. We provide an upper bound for $\gamma_{\stackrel{}{1k}}(T)$ in any tree $T$ and trees achieving this bound are characterized. We prove that there exist trees satisfying all the possible equalities and inequalities in this chain and a linear algorithm for computing $\gamma_{\stackrel{}{1k}}(T)$ in any tree is presented.  [Show abstract] [Hide abstract]
ABSTRACT: A subset $S\subseteq V$ in a graph $G=(V,E)$ is a $k$quasiperfect dominating set (for $k\geq 1$) if every vertex not in $S$ is adjacent to at least one and at most $k$ vertices in $S$. The cardinality of a minimum $k$quasiperfect dominating set in $G$ is denoted by $\gamma_ {\stackrel{}{1k}}(G)$. Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers $ n \ge \gamma_ {\stackrel{}{11}}(G) \ge \gamma_ {\stackrel{}{12}}(G)\ge \ldots \ge \gamma_ {\stackrel{}{1\Delta}}(G)=\gamma(G)$ in order to indicate how far is $G$ from being perfectly dominated. In this paper we study properties, existence and realization of graphs for which the chain is short, that is, $\gamma_ {\stackrel{}{12}}(G)=\gamma (G)$. Among them, one can find cographs, clawfree graphs and graphs with extremal values of $\Delta(G)$. 
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ABSTRACT: In the graph distance game, two players alternate in constructing a maximal path. The objective function is the distance between the two endpoints of the path, which one player tries to maximize and the other tries to minimize. In this paper we examine the distance game for various graph operations: the join, the corona and the lexicographic product of graphs. We provide general bounds and exact results for special graphs.  [Show abstract] [Hide abstract]
ABSTRACT: Given a graph $G$ and a vertex $x\in V(G)$, a vertex set $S \subseteq V(G)$ is an $x$geodominating set of $G$ if each vertex $v\in V(G)$ lies on an $xy$ geodesic for some element $y\in S$. The minimum cardinality of an $x$geodominating set of $G$ is defined as the $x$geodomination number of $G$, $g_x(G)$, and an $x$geodominating set of cardinality $g_x(G)$ is called a $g_x$set and it is known that it is unique for each vertex $x$. We prove that, in any graph $G$, the $g_x$set associated to a vertex $x$ is the set of boundary vertices of $x$, that is $\partial(x)= \{v \in V(G) : \forall w \in N(v): d(x,w) \leq d(u, v)\}$. This characterization of $g_x$sets allows to deduce, on a easy way, different properties of these sets and also to compute both $g_x$sets and $x$geodomination number $g_x(G)$, in graphs obtained using different graphs products: cartesian, strong and lexicographic. 
Article: Graph operations and Lie algebras
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ABSTRACT: This paper deals with several operations on graphs and combinatorial structures linking them with their associated Lie algebras. More concretely, our main goal is to obtain some criteria to determine when there exists a Lie algebra associated with a combinatorial structure arising from those operations. Additionally, we show an algorithmic method for one of those operations. 
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ABSTRACT: A set of vertices S is a determining set of a graph G if every automorphism of G is uniquely determined by its action on S. The determining number of G is the minimum cardinality of a determining set of G. This paper studies the determining number of Kneser graphs. First, we compute the determining number of a wide range of Kneser graphs, concretely K n:k with n≥k(k+1)/2+1. In the language of group theory, these computations provide exact values for the base size of the symmetric group S n acting on the ksubsets of {1,⋯,n}. Then, we establish for which Kneser graphs K n:k the determining number is equal to nk, answering a question posed by Boutin. Finally, we find all Kneser graphs with fixed determining number 5, extending the study developed by Boutin for determining number 2, 3 or 4.  [Show abstract] [Hide abstract]
ABSTRACT: A set of vertices SSresolves a graph GG if every vertex is uniquely determined by its vector of distances to the vertices in SS. The metric dimension of a graph GG is the minimum cardinality of a resolving set. In this paper we study the metric dimension of infinite graphs such that all its vertices have finite degree. We give necessary conditions for those graphs to have finite metric dimension and characterize infinite trees with finite metric dimension. We also establish some results about the metric dimension of the cartesian product of finite and infinite graphs, and give the metric dimension of the cartesian product of several families of graphs.  [Show abstract] [Hide abstract]
ABSTRACT: This paper shows a characterization of digraphs of three vertices associated with Lie algebras, as well as determining the list of isomorphism classes for Lie algebras associated with these digraphs. Additionally, we introduce and implement two algorithmic procedures related to this study: the first is devoted to draw, if exists, the digraph associated with a given Lie algebra; whereas the other corresponds to the converse problem and allows us to test if a given digraph is associated or not with a Lie algebra. Finally, we give the complete list of all nonisomorphic combinatorial structures of three vertices associated with Lie algebras and we study the type of Lie algebra associated with each configuration.  [Show abstract] [Hide abstract]
ABSTRACT: In this work, two types of codes such that they both dominate and locate the vertices of a graph are studied. Those codes might be sets of detectors in a network or processors controlling a system whose set of responses should determine a malfunctioning processor or an intruder. Here, we present our more significant contributions on \lambdacodes and \etacodes concerning concerning bounds, extremal values and realization theorems.  [Show abstract] [Hide abstract]
ABSTRACT: A set of vertices $S$ in a graph $G$ is a {\em resolving set} for $G$ if, for any two vertices $u,v$, there exists $x\in S$ such that the distances $d(u,x) \neq d(v,x)$. In this paper, we consider the Johnson graphs $J(n,k)$ and Kneser graphs $K(n,k)$, and obtain various constructions of resolving sets for these graphs. As well as general constructions, we show that various interesting combinatorial objects can be used to obtain resolving sets in these graphs, including (for Johnson graphs) projective planes and symmetric designs, as well as (for Kneser graphs) partial geometries, Hadamard matrices, Steiner systems and toroidal grids.  [Show abstract] [Hide abstract]
ABSTRACT: Let V be a finite set and M a collection of subsets of V. Then M is an alignment of V if and only if M is closed under taking intersections and contains both V and the empty set. If M is an alignment of V, then the elements of M are called convex sets and the pair (V,M) is called an alignment or a convexity. If S⊆V, then the convex hull of S is the smallest convex set that contains S. Suppose X∈M. Then x∈X is an extreme point for X if X∖{x}∈M. A convex geometry on a finite set is an aligned space with the additional property that every convex set is the convex hull of its extreme points. Let G=(V,E) be a connected graph and U a set of vertices of G. A subgraph T of G containing U is a minimal Utree if T is a tree and if every vertex of V(T)∖U is a cutvertex of the subgraph induced by V(T). The monophonic interval of U is the collection of all vertices of G that belong to some minimal Utree. Several graph convexities are defined using minimal Utrees and structural characterizations of graph classes for which the corresponding collection of convex sets is a convex geometry are characterized.  [Show abstract] [Hide abstract]
ABSTRACT: A set of vertices $S$ is a \emph{determining set} of a graph $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The \emph{determining number} of $G$ is the minimum cardinality of a determining set of $G$. This paper studies determining sets of Kneser graphs from a hypergraph perspective. This new technique lets us compute the determining number of a wide range of Kneser graphs, concretely $K_{n:k}$ with $n\geq \frac{k(k+1)}{2}+1$. We also show its usefulness by giving shorter proofs of the characterization of all Kneser graphs with fixed determining number 2, 3 or 4, going even further to fixed determining number 5. We finally establish for which Kneser graphs $K_{n:k}$ the determining number is equal to $nk$, answering a question posed by Boutin. 
Article: m_3^3Convex geometries are Afree
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ABSTRACT: Let V be a finite set and M a collection of subsets of V. Then M is an alignment of V if and only if M is closed under taking intersections and contains both V and the empty set. If M is an alignment of V, then the elements of M are called convex sets and the pair (V, M) is called an aligned space. If S is a subset of V, then the convex hull of S is the smallest convex set that contains S. Suppose X in M. Then x in X is an extreme point for X if Xx is in M. The collection of all extreme points of X is denoted by ex(X). A convex geometry on a finite set is an aligned space with the additional property that every convex set is the convex hull of its extreme points. Let G=(V,E) be a connected graph and U a set of vertices of G. A subgraph T of G containing U is a minimal Utree if T is a tree and if every vertex of V(T)U is a cutvertex of the subgraph induced by V(T). The monophonic interval of U is the collection of all vertices of G that belong to some minimal Utree. A set S of vertices in a graph is m_kconvex if it contains the monophonic interval of every kset of vertices is S. A set of vertices S of a graph is m^3convex if for every pair u,v of vertices in S, the vertices on every induced path of length at least 3 are contained in S. A set S is m_3^3convex if it is both m_3 and m^3 convex. We show that if the m_3^3convex sets form a convex geometry, then G is Afree.  [Show abstract] [Hide abstract]
ABSTRACT: Dominating concepts constitute a cornerstone in Graph Theory. Part of the efforts in the field have been focused in finding different mathematical frameworks where domination notions naturally arise, providing new points of view about the matter. In this paper, we introduce one of these frameworks based in convexity. The main idea consists of defining a convexity in a graph, already used in image processing, for which the usual parameters of convexity are closely related to domination parameters. Moreover, the Helly number of this convexity may be viewed as a new domination parameter whose study would be of interest.  [Show abstract] [Hide abstract]
ABSTRACT: Postprint (published version)  [Show abstract] [Hide abstract]
ABSTRACT: Preprint  [Show abstract] [Hide abstract]
ABSTRACT: This paper initiates a study on the problem of computing the difference between the metric dimension and the determining number of graphs. We provide new proofs and results on the determining number of trees and Cartesian products of graphs, and establish some lower bounds on the difference between the two parameters. Postprint (published version)
Publication Stats
449  Citations  
18.90  Total Impact Points  
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Institutions

19982015

Universidad de Almería
 Department of Mathematics
Unci, Andalusia, Spain 
University of Helsinki
 Department of Mathematics and Statistics
Helsinki, Uusimaa, Finland
