8,629 research outputs found
Client–Server and Cost Effective Sets in Graphs
For any integer k≥0, a set of vertices S of a graph G=(V,E) is k-cost-effective if for every v∈S,|N(v)∩(V∖S)|≥|N(v)∩S|+k. In this paper we study the minimum cardinality of a maximal k-cost-effective set and the maximum cardinality of a k-cost-effective set. We obtain Gallai-type results involving the k-cost-effective and global k-offensive alliance parameters, and we provide bounds on the maximum k-cost-effective number. Finally, we consider k-cost-effective sets that are also dominating. We show that computing the k-cost-effective domination number is NP-complete for bipartite graphs. Moreover, we note that not all trees have a k-cost-effective dominating set and give a constructive characterization of those that do
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 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
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
Open k-monopolies in graphs: complexity and related concepts
Closed monopolies in graphs have a quite long range of applications in
several problems related to overcoming failures, since they frequently have
some common approaches around the notion of majorities, for instance to
consensus problems, diagnosis problems or voting systems. We introduce here
open -monopolies in graphs which are closely related to different parameters
in graphs. Given a graph and , if is the
number of neighbors has in , is an integer and is a positive
integer, then we establish in this article a connection between the following
three concepts:
- Given a nonempty set a vertex of is said to be
-controlled by if . The set
is called an open -monopoly for if it -controls every vertex of
.
- A function is called a signed total
-dominating function for if for all
.
- A nonempty set is a global (defensive and offensive)
-alliance in if holds for every .
In this article we prove that the problem of computing the minimum
cardinality of an open -monopoly in a graph is NP-complete even restricted
to bipartite or chordal graphs. In addition we present some general bounds for
the minimum cardinality of open -monopolies and we derive some exact values.Comment: 18 pages, Discrete Mathematics & Theoretical Computer Science (2016
Alliance free and alliance cover sets
A \emph{defensive} (\emph{offensive}) -\emph{alliance} in
is a set such that every in (in the boundary of ) has
at least more neighbors in than it has in . A set
is \emph{defensive} (\emph{offensive}) -\emph{alliance free,}
if for all defensive (offensive) -alliance , ,
i.e., does not contain any defensive (offensive) -alliance as a subset.
A set is a \emph{defensive} (\emph{offensive})
-\emph{alliance cover}, if for all defensive (offensive) -alliance ,
, i.e., contains at least one vertex from each
defensive (offensive) -alliance of . In this paper we show several
mathematical properties of defensive (offensive) -alliance free sets and
defensive (offensive) -alliance cover sets, including tight bounds on the
cardinality of defensive (offensive) -alliance free (cover) sets
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