4,176 research outputs found
On the complement graph and defensive k-alliances
AbstractIn this paper, we obtain several tight bounds of the defensive k-alliance number in the complement graph from other parameters of the graph. In particular, we investigate the relationship between the alliance numbers of the complement graph and the minimum and maximum degree, the domination number and the isoperimetric number of the graph. Moreover, we prove the NP-completeness of the decision problem underlying the defensive k-alliance number
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
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
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
- …