20,433 research outputs found
On global location-domination in graphs
A dominating set of a graph is called locating-dominating, LD-set for
short, if every vertex not in is uniquely determined by the set of
neighbors of belonging to . Locating-dominating sets of minimum
cardinality are called -codes and the cardinality of an LD-code is the
location-domination number . An LD-set of a graph is global
if it is an LD-set of both and its complement . The global
location-domination number is the minimum cardinality of a
global LD-set of . In this work, we give some relations between
locating-dominating sets and the location-domination number in a graph and its
complement.Comment: 15 pages: 2 tables; 8 figures; 20 reference
On global location-domination in graphs
A dominating set S of a graph G is called locating-dominating, LD-set for short, if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S. Locating-dominating sets of minimum cardinality are called LD-codes and the cardinality of an LD-code is the location-domination number lambda(G). An LD-set S of a graph G is global if it is an LD-set of both G and its complement G'. The global location-domination number lambda g(G) is introduced as the minimum cardinality of a global LD-set of G.
In this paper, some general relations between LD-codes and the location-domination number in a graph and its complement are presented first.
Next, a number of basic properties involving the global location-domination number are showed. Finally, both parameters are studied in-depth for the family of block-cactus graphs.Postprint (published version
Solving Two Conjectures regarding Codes for Location in Circulant Graphs
Identifying and locating-dominating codes have been widely studied in
circulant graphs of type , which can also be viewed as
power graphs of cycles. Recently, Ghebleh and Niepel (2013) considered
identification and location-domination in the circulant graphs . They
showed that the smallest cardinality of a locating-dominating code in
is at least and at most
for all . Moreover, they proved that the lower bound is strict when
and conjectured that the lower bound can be
increased by one for other . In this paper, we prove their conjecture.
Similarly, they showed that the smallest cardinality of an identifying code in
is at least and at most for all . Furthermore, they proved that the lower bound is
attained for most of the lengths and conjectured that in the rest of the
cases the lower bound can improved by one. This conjecture is also proved in
the paper. The proofs of the conjectures are based on a novel approach which,
instead of making use of the local properties of the graphs as is usual to
identification and location-domination, also manages to take advantage of the
global properties of the codes and the underlying graphs
Solving Two Conjectures regarding Codes for Location in Circulant Graphs
Identifying and locating-dominating codes have been widely studied in circulant graphs of type Cn(1, 2, . . ., r), which can also be viewed as power graphs of cycles. Recently, Ghebleh and Niepel (2013) considered identification and location-domination in the circulant graphs Cn(1, 3). They showed that the smallest cardinality of a locating-dominating code in Cn(1, 3) is at least ⌈n/3⌉ and at most ⌈n/3⌉ + 1 for all n ≥ 9. Moreover, they proved that the lower bound is strict when n ≡ 0, 1, 4 (mod 6) and conjectured that the lower bound can be increased by one for other n. In this paper, we prove their conjecture. Similarly, they showed that the smallest cardinality of an identifying code in Cn(1, 3) is at least ⌈4n/11⌉ and at most ⌈4n/11⌉ + 1 for all n ≥ 11. Furthermore, they proved that the lower bound is attained for most of the lengths n and conjectured that in the rest of the cases the lower bound can improved by one. This conjecture is also proved in the paper. The proofs of the conjectures are based on a novel approach which, instead of making use of the local properties of the graphs as is usual to identification and location-domination, also manages to take advantage of the global properties of the codes and the underlying graphs.</p
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
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
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