12 research outputs found
Variant Domination Types for a Complete h-ary Tree
يعتبر البيان أداة جيدة لحلول بعض مشاكل الشبكات. من هذة المشاكل هي مسالة الهيمنة في الشبكات والتي تدرس عن طريق نظرية البيانات بعد تحويل الشبكة الي بيان الذي هو مجموعة من الرؤوس مع مجموعة من الحافات التي تربط بين هذه الرؤوس. اي مجموعة جزيئة من رؤوس البيان هي مجموعة هيمنة في البيان اذا كان اي راس في البيان اما ينتمي لمجموعة الهيمنة او له جوار في هذه المجموعة. رقم الهيمنة هو قياس اصغر مجموعة تهيمن على البيان. لأهمية هذا الموضوع في مختلف المجالات ، انواع مختلف من الهيمنة في البيانات تم استخدامها في هذا البحث. وذلك بوضع شرط على مجموعة الهيمنة او شرط على مجموعة باقي رؤوس البيان او على المجموعتين. في هذا البحث تم اختيار نوعين من الهيمنة . الاول هو الهيمنة المقيدة حيث الشرط وضع على مجموعة الرؤوس خارج المجموعة المقيدة . اما النوع الثاني وهو الهيمنة الآمنة وهي المجموعة المهيمنة مع وضع شرط على المجموعة المهيمنة. تم دراسة الهيمنة الآمنة مع انواع مختلفة من الهيمنة المقيدة على عائلة من الاشجاروهي اشجار شعاع الجذر المتكامل ذو العمق .Graph is a tool that can be used to simplify and solve network problems. Domination is a typical network problem that graph theory is well suited for. A subset of nodes in any network is called dominating if every node is contained in this subset, or is connected to a node in it via an edge. Because of the importance of domination in different areas, variant types of domination have been introduced according to the purpose they are used for. In this paper, two domination parameters the first is the restrained and the second is secure domination have been chosn. The secure domination, and some types of restrained domination in one type of trees is called complete ary tree are determined
Secure domination number of -subdivision of graphs
Let be a simple graph. A dominating set of is a subset
such that every vertex not in is adjacent to at least one
vertex in . The cardinality of a smallest dominating set of , denoted by
, is the domination number of . A dominating set is called a
secure dominating set of , if for every , there exists a vertex
such that and is a dominating set of
. The cardinality of a smallest secure dominating set of , denoted by
, is the secure domination number of . For any , the -subdivision of is a simple graph
which is constructed by replacing each edge of with a path of length .
In this paper, we study the secure domination number of -subdivision of .Comment: 10 Pages, 8 Figure
Algorithmic Complexity of Isolate Secure Domination in Graphs
A dominating set is an Isolate Dominating Set (IDS) if the induced
subgraph has at least one isolated vertex. In this paper, we initiate
the study of new domination parameter called, isolate secure domination. An
isolate dominating set is an isolate secure dominating set
(ISDS), if for each vertex , there exists a neighboring
vertex of in such that is an IDS
of . The minimum cardinality of an ISDS of is called as an isolate
secure domination number, and is denoted by . Given a graph and a positive integer the ISDM problem is to check whether has an isolate secure dominating set of size at most We prove that ISDM
is NP-complete even when restricted to bipartite graphs and split graphs. We
also show that ISDM can be solved in linear time for graphs of bounded
tree-width.Comment: arXiv admin note: substantial text overlap with arXiv:2002.00002;
text overlap with arXiv:2001.1125
Algorithmic complexity of isolate secure domination in graphs
A dominating set S is an Isolate Dominating Set (IDS) if the induced subgraph G[S] has at least one isolated vertex. In this paper, we initiate the study of new domination parameter called, isolate secure domination. An isolate dominating set S subset of V is an isolate secure dominating set (ISDS), if for each vertex u is an element of V \ S, there exists a neighboring vertex v of u in S such that (S \ {v}) boolean OR {u} is an IDS of G. The minimum cardinality of an ISDS of G is called as an isolate secure domination number, and is denoted by gamma(0s) (G). We give isolate secure domination number of path and cycle graphs. Given a graph G = (V, E) and a positive integer k, the ISDM problem is to check whether G has an isolate secure dominating set of size at most k. We prove that ISDM is NP-complete even when restricted to bipartite graphs and split graphs. We also show that ISDM can be solved in linear time for graphs of bounded tree-width.Publisher's Versio
On the Complexity of Co-secure Dominating Set Problem
A set of a graph is a dominating set of if
every vertex is adjacent to at least one vertex in A
set is a co-secure dominating set (CSDS) of a graph if
is a dominating set of and for each vertex there exists a vertex
such that and
is a dominating set of . The minimum cardinality of a co-secure dominating
set of is the co-secure domination number and it is denoted by
. Given a graph , the minimum co-secure dominating
set problem (Min Co-secure Dom) is to find a co-secure dominating set of
minimum cardinality. In this paper, we strengthen the inapproximability result
of Min Co-secure Dom for general graphs by showing that this problem can not be
approximated within a factor of for perfect elimination
bipartite graphs and star convex bipartite graphs unless P=NP. On the positive
side, we show that Min Co-secure Dom can be approximated within a factor of
for any graph with . For -regular and
-regular graphs, we show that Min Co-secure Dom is approximable within a
factor of and , respectively. Furthermore, we
prove that Min Co-secure Dom is APX-complete for -regular graphs.Comment: 12 pages, 2 figure
Algorithmic Aspects of Secure Connected Domination in Graphs
Let be a simple, undirected and connected graph. A connected
dominating set is a secure connected dominating set of , if
for each , there exists such that
and the set is a connected dominating set
of . The minimum size of a secure connected dominating set of denoted by
, is called the secure connected domination number of .
Given a graph and a positive integer the Secure Connected Domination
(SCDM) problem is to check whether has a secure connected dominating set
of size at most In this paper, we prove that the SCDM problem is
NP-complete for doubly chordal graphs, a subclass of chordal graphs. We
investigate the complexity of this problem for some subclasses of bipartite
graphs namely, star convex bipartite, comb convex bipartite, chordal bipartite
and chain graphs. The Minimum Secure Connected Dominating Set (MSCDS) problem
is to find a secure connected dominating set of minimum size in the input
graph. We propose a - approximation algorithm for MSCDS,
where is the maximum degree of the input graph and prove
that MSCDS cannot be approximated within for any unless even for
bipartite graphs. Finally, we show that the MSCDS is APX-complete for graphs
with
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 guàrdia, 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 mínim
d'entitats necessàries per a la protecció sota cada estratègia és d'interès. En general, s'obtenen fórmules tancades o
fites ajustades sobre els paràmetres 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
teoría de la dominación en grafos, o en la teoría 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 artículos, 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 mínimo 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