5 research outputs found
Global offensive -alliances in digraphs
In this paper, we initiate the study of global offensive -alliances in
digraphs. Given a digraph , a global offensive -alliance in a
digraph is a subset such that every vertex outside of
has at least one in-neighbor from and also at least more in-neighbors
from than from outside of , by assuming is an integer lying between
two minus the maximum in-degree of and the maximum in-degree of . The
global offensive -alliance number is the minimum
cardinality among all global offensive -alliances in . In this article we
begin the study of the global offensive -alliance number of digraphs. For
instance, we prove that finding the global offensive -alliance number of
digraphs is an NP-hard problem for any value and that it remains NP-complete even when
restricted to bipartite digraphs when we consider the non-negative values of
given in the interval above. Based on these facts, lower bounds on
with characterizations of all digraphs attaining the bounds
are given in this work. We also bound this parameter for bipartite digraphs
from above. For the particular case , an immediate result from the
definition shows that for all digraphs ,
in which stands for the domination number of . We show that
these two digraph parameters are the same for some infinite families of
digraphs like rooted trees and contrafunctional digraphs. Moreover, we show
that the difference between and can be
arbitrary large for directed trees and connected functional digraphs
Alliance free sets in Cartesian product graphs
Let be a graph. For a non-empty subset of vertices ,
and vertex , let denote the
cardinality of the set of neighbors of in , and let .
Consider the following condition: {equation}\label{alliancecondition}
\delta_S(v)\ge \delta_{\bar{S}}(v)+k, \{equation} which states that a vertex
has at least more neighbors in than it has in . A set
that satisfies Condition (\ref{alliancecondition}) for every
vertex is called a \emph{defensive} -\emph{alliance}; for every
vertex in the neighborhood of is called an \emph{offensive}
-\emph{alliance}. A subset of vertices , is a \emph{powerful}
-\emph{alliance} if it is both a defensive -alliance and an offensive -alliance. Moreover, a subset is a defensive (an offensive or
a powerful) -alliance free set if does not contain any defensive
(offensive or powerful, respectively) -alliance. In this article we study
the relationships between defensive (offensive, powerful) -alliance free
sets in Cartesian product graphs and defensive (offensive, powerful)
-alliance free sets in the factor graphs
On global offensive k-alliances in graphs
We investigate the relationship between global offensive k-alliances and some characteristic sets of a graph including r-dependent sets, Ï„-dominating sets and standard dominating sets. In addition, we discuss the close relationship that exist among the (global) offensive ki-alliance number of Γi, i ∈ {1,2} and the (global) offensive k-alliance number of Γ1×Γ2, for some specific values of k. As a consequence of the study, we obtain bounds on the global offensive k-alliance number in terms of several parameters of the graph. e-mail:[email protected]. Partially supported by Ministerio de Ciencia y TecnologÃa, ref. BFM2003-00034 and Junta de AndalucÃa, ref. FQM-260 and ref. P06-FQM-02225