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Offensive alliances in cubic graphs
An offensive alliance in a graph is a set of vertices
where for every vertex in its boundary it holds that the
majority of vertices in 's closed neighborhood are in . In the case of
strong offensive alliance, strict majority is required. An alliance is
called global if it affects every vertex in , that is, is a
dominating set of . The global offensive alliance number
(respectively, global strong offensive alliance number
) is the minimum cardinality of a global offensive
(respectively, global strong offensive) alliance in . If has
global independent offensive alliances, then the \emph{global independent
offensive alliance number} is the minimum cardinality among
all independent global offensive alliances of . In this paper we study
mathematical properties of the global (strong) alliance number of cubic graphs.
For instance, we show that for all connected cubic graph of order ,
where
denotes the line graph of . All the above bounds are tight
Global defensive k-alliances in graphs
Let be a simple graph. For a nonempty set , and
a vertex , denotes the number of neighbors has in
. A nonempty set is a \emph{defensive -alliance} in
if A
defensive -alliance is called \emph{global} if it forms a dominating
set. The \emph{global defensive -alliance number} of , denoted by
, is the minimum cardinality of a defensive
-alliance in . We study the mathematical properties of
Defensive alliances in graphs: a survey
A set of vertices of a graph is a defensive -alliance in if
every vertex of has at least more neighbors inside of than outside.
This is primarily an expository article surveying the principal known results
on defensive alliances in graph. Its seven sections are: Introduction,
Computational complexity and realizability, Defensive -alliance number,
Boundary defensive -alliances, Defensive alliances in Cartesian product
graphs, Partitioning a graph into defensive -alliances, and Defensive
-alliance free sets.Comment: 25 page
A Survey on Alliances and Related Parameters in Graphs
In this paper, we show that several graph parameters are known in different areas under completely different names.More specifically, our observations connect signed domination, monopolies, -domination, -independence,positive influence domination,and a parameter associated to fast information propagationin networks to parameters related to various notions of global -alliances in graphs.We also propose a new framework, called (global) -alliances, not only in order to characterizevarious known variants of alliance and domination parameters, but also to suggest a unifying framework for the study of alliances and domination.Finally, we also give a survey on the mentioned graph parameters, indicating how results transfer due to our observations
Client–server and cost effective sets in graphs
Abstract: Please refer to full text to view abstract
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