[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 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. 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, claw-free
graphs and graphs with extremal values of $\Delta(G)$.
[Show abstract][Hide abstract] 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.
Electronic Notes in Discrete Mathematics 09/2014; 46(1):153–159. DOI:10.1016/j.endm.2014.08.021
[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 $x-y$
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.
[Show abstract][Hide abstract] 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.
International Journal of Computer Mathematics 10/2013; 90(10):2092-2104. DOI:10.1080/00207160.2013.777048 · 0.82 Impact Factor
[Show abstract][Hide abstract] 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 k-subsets of {1,⋯,n}. Then, we establish for which Kneser graphs K n:k the determining number is equal to n-k, 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 non-isomorphic combinatorial structures of three vertices associated with Lie algebras
and we study the type of Lie algebra associated with each configuration.
International Journal of Computer Mathematics 09/2012; 89(13-14):1879-1900. DOI:10.1080/00207160.2012.688114 · 0.82 Impact Factor
[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 \lambda-codes and \eta-codes 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.
European Journal of Combinatorics 03/2012; 34(4). DOI:10.1016/j.ejc.2012.10.008 · 0.65 Impact Factor
[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 U-tree if T is a tree and if every vertex of V(T)∖U is a cut-vertex 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 U-tree. Several graph convexities are defined using minimal U-trees 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 $n-k$,
answering a question posed by Boutin.
[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 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 X-x 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 U-tree if T is a tree and if every vertex of V(T)-U is a cut-vertex
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 U-tree. A set S of vertices in
a graph is m_k-convex if it contains the monophonic interval of every k-set of
vertices is S. A set of vertices S of a graph is m^3-convex 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_3^3-convex if it is both m_3- and m^3- convex.
We show that if the m_3^3-convex sets form a convex geometry, then G is A-free.
[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: 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)
Electronic Journal of Combinatorics 12/2010; 17(1). · 0.49 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: A set of vertices of a connected graph is convex, if for any pair of vertices , every shortest path joining and is contained in . The convex hull of a set of vertices is defined as the smallest convex set in containing . The set is geodetic, if every vertex of lies on some shortest path joining two vertices in , and it is said to be a hull set if its convex hull is . The geodetic and the hull numbers of are the minimum cardinality of a 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 establish 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.