24,147 research outputs found
On the extremal properties of the average eccentricity
The eccentricity of a vertex is the maximum distance from it to another
vertex and the average eccentricity of a graph is the mean value
of eccentricities of all vertices of . The average eccentricity is deeply
connected with a topological descriptor called the eccentric connectivity
index, defined as a sum of products of vertex degrees and eccentricities. In
this paper we analyze extremal properties of the average eccentricity,
introducing two graph transformations that increase or decrease .
Furthermore, we resolve four conjectures, obtained by the system AutoGraphiX,
about the average eccentricity and other graph parameters (the clique number,
the Randi\' c index and the independence number), refute one AutoGraphiX
conjecture about the average eccentricity and the minimum vertex degree and
correct one AutoGraphiX conjecture about the domination number.Comment: 15 pages, 3 figure
(Total) Vector Domination for Graphs with Bounded Branchwidth
Given a graph of order and an -dimensional non-negative
vector , called demand vector, the vector domination
(resp., total vector domination) is the problem of finding a minimum
such that every vertex in (resp., in ) has
at least neighbors in . The (total) vector domination is a
generalization of many dominating set type problems, e.g., the dominating set
problem, the -tuple dominating set problem (this is different from the
solution size), and so on, and its approximability and inapproximability have
been studied under this general framework. In this paper, we show that a
(total) vector domination of graphs with bounded branchwidth can be solved in
polynomial time. This implies that the problem is polynomially solvable also
for graphs with bounded treewidth. Consequently, the (total) vector domination
problem for a planar graph is subexponential fixed-parameter tractable with
respectto , where is the size of solution.Comment: 16 page
Computational complexity aspects of super domination
Let G be a graph. A dominating set D ⊆ V (G) is a super dominating set if for every vertex x ∈ V (G) \ D there exists y ∈ D such that NG (y) ∩ (V (G) \ D)) = {x}. The cardinality of a smallest super dominating set of G is the super domination number of G. An exact formula for the super domination number of a tree T is obtained, and it is demonstrated that a smallest super dominating set of T can be computed in linear time. It is proved that it is NP-complete to decide whether the super domination number of a graph G is at most a given integer if G is a bipartite graph of girth at least 8. The super domination number is determined for all k-subdivisions of graphs. Interestingly, in half of the cases the exact value can be efficiently computed from the obtained formulas, while in the other cases the computation is hard. While obtaining these formulas, II-matching numbers are introduced and proved that they are computationally hard to determinepublishedVersio
Vertex-Edge and Edge-Vertex Parameters in Graphs
The majority of graph theory research on parameters involved with domination, independence, and irredundance has focused on either sets of vertices or sets of edges; for example, sets of vertices that dominate all other vertices or sets of edges that dominate all other edges. There has been very little research on ``mixing\u27\u27 vertices and edges. We investigate several new and several little-studied parameters, including vertex-edge domination, vertex-edge irredundance, vertex-edge independence, edge-vertex domination, edge-vertex irredundance, and edge-vertex independence
Total protection in graphs
Suposem que una o diverses entitats estan situades en alguns dels vèrtexs d'un graf simple, i que una entitat situada
en un vèrtex es pot ocupar d'un problema en qualsevol vèrtex del seu entorn tancat. En general, una entitat pot
consistir en un robot, un observador, una legió, un guà rdia, etc. Informalment, diem que un graf està protegit sota una
determinada ubicació d'entitats si hi ha almenys una entitat disponible per tractar un problema en qualsevol vèrtex.
S'han considerat diverses estratègies (o regles d'ubicació d'entitats), sota cadascuna de les quals el graf es considera
protegit. Aquestes estratègies de protecció de grafs s'emmarquen en la teoria de la dominació en grafs, o en la teoria
de la dominació segura en grafs.
En aquesta tesi, introduïm l'estudi de la w-dominació (segura) en grafs, el qual és un enfocament unificat a la idea de
protecció de grafs, i que engloba variants conegudes de dominació (segura) en grafs i introdueix de noves.
La tesi està estructurada com un compendi de deu articles, els quals han estat publicats en revistes indexades en el
JCR. El primer està dedicat a l'estudi de la w-dominació, el cinquè a l'estudi de la w-dominació segura, mentre que els
altres treballs estan dedicats a casos particulars d'estratègies de protecció total. Com és d'esperar, el nombre mÃnim
d'entitats necessà ries per a la protecció sota cada estratègia és d'interès. En general, s'obtenen fórmules tancades o
fites ajustades sobre els parà metres estudiats.Supongamos que una o varias entidades están situadas en algunos de los vértices de un grafo simple y que una
entidad situada en un vértice puede ocuparse de un problema en cualquier vértice de su vecindad cerrada. En general,
una entidad puede consistir en un robot, un observador, una legión, un guardia, etc. Informalmente, decimos que un
grafo está protegido bajo una determinada ubicación de entidades si existe al menos una entidad disponible para tratar
un problema en cualquier vértice. Se han considerado varias estrategias (o reglas de ubicación de entidades), bajo
cada una de las cuales el grafo se considera protegido. Estas estrategias de protección de grafos se enmarcan en la
teorÃa de la dominación en grafos, o en la teorÃa de la dominación segura en grafos.
En esta tesis, introducimos el estudio de la w-dominación (segura) en grafos, el cual es un enfoque unificado a la idea
de protección de grafos, y que engloba variantes conocidas de dominación (segura) en grafos e introduce otras
nuevas. La tesis está estructurada como un compendio de diez artÃculos, los cuales han sido publicados en revistas
indexadas en el JCR. El primero está dedicado al estudio de la w-dominación, el quinto al estudio de la w-dominación
segura, mientras que los demás trabajos están dedicados a casos particulares de estrategias de protección total.
Como es de esperar, el número mÃnimo de entidades necesarias para la protección bajo cada estrategia es de interés.
En general, se obtienen fórmulas cerradas o cotas ajustadas sobre los parámetros estudiadosSuppose that one or more entities are stationed at some of the vertices of a simple graph and that an entity at a vertex
can deal with a problem at any vertex in its closed neighbourhood. In general, an entity could consist of a robot, an
observer, a legion, a guard, and so on. Informally, we say that a graph is protected under a given placement of
entities if there exists at least one entity available to handle a problem at any vertex. Various strategies (or rules for
entities placements) have been considered, under each of which the graph is deemed protected. These strategies for
the protection of graphs are framed within the theory of domination in graphs, or in the theory of secure domination in
graphs.
In this thesis, we introduce the study of (secure) w-domination in graphs, which is a unified approach to the idea of
protection of graphs, that encompasses known variants of (secure) domination in graphs and introduces new ones.
The thesis is structured as a compendium of ten papers which have been published in JCR-indexed journals. The first
one is devoted to the study of w-domination, the fifth one is devoted to the study of secure w-domination, while the
other papers are devoted to particular cases of total protection strategies. As we can expect, the minimum number of
entities required for protection under each strategy is of interest. In general, we obtain closed formulas or tight bounds
on the studied parameters
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