6,985 research outputs found
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
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
New bounds on the signed total domination number of graphs
In this paper, we study the signed total domination number in graphs and
present new sharp lower and upper bounds for this parameter. For example by
making use of the classic theorem of Turan, we present a sharp lower bound on
this parameter for graphs with no complete graph of order r+1 as a subgraph.
Also, we prove that n-2(s-s') is an upper bound on the signed total domination
number of any tree of order n with s support vertices and s' support vertives
of degree two. Moreover, we characterize all trees attainig this bound.Comment: This paper contains 11 pages and one figur
Protecting a Graph with Mobile Guards
Mobile guards on the vertices of a graph are used to defend it against
attacks on either its vertices or its edges. Various models for this problem
have been proposed. In this survey we describe a number of these models with
particular attention to the case when the attack sequence is infinitely long
and the guards must induce some particular configuration before each attack,
such as a dominating set or a vertex cover. Results from the literature
concerning the number of guards needed to successfully defend a graph in each
of these problems are surveyed.Comment: 29 pages, two figures, surve
Some Results on incidence coloring, star arboricity and domination number
Two inequalities bridging the three isolated graph invariants, incidence
chromatic number, star arboricity and domination number, were established.
Consequently, we deduced an upper bound and a lower bound of the incidence
chromatic number for all graphs. Using these bounds, we further reduced the
upper bound of the incidence chromatic number of planar graphs and showed that
cubic graphs with orders not divisible by four are not 4-incidence colorable.
The incidence chromatic numbers of Cartesian product, join and union of graphs
were also determined.Comment: 8 page
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