127 research outputs found
The Hosoya polynomial of distance-regular graphs
In this note we obtain an explicit formula for the Hosoya polynomial of any
distance-regular graph in terms of its intersection array. As a consequence, we
obtain a very simple formula for the Hosoya polynomial of any strongly regular
graph
Alliances and related parameters in graphs
In this paper, we show that several graph parameters are known in different
areas under completely different names. More specifically, our observations
connect signed domination, monopolies, -domination,
-independence, positive influence domination, and a parameter
associated to fast information propagation in networks to parameters related to
various notions of global -alliances in graphs. We also propose a new
framework, called (global) -alliances, not only in order to characterize
various known variants of alliance and domination parameters, but also to
suggest a unifying framework for the study of alliances and domination.
Finally, we also give a survey on the mentioned graph parameters, indicating
how results transfer due to our observations
The local metric dimension of strong product graphs
A vertex is said to distinguish two vertices of a
nontrivial connected graph if the distance from to is different
from the distance from to . A set is a local metric
generator for if every two adjacent vertices of are distinguished by
some vertex of . A local metric generator with the minimum cardinality is
called a local metric basis for and its cardinality, the local metric
dimension of . It is known that the problem of computing the local metric
dimension of a graph is NP-Complete. In this paper we study the problem of
finding exact values or bounds for the local metric dimension of strong product
of graphs
Roman domination in Cartesian product graphs and strong product graphs
A set of vertices of a graph is a dominating set for if every
vertex outside of is adjacent to at least one vertex belonging to . The
minimum cardinality of a dominating set for is called the domination number
of . A map is a Roman dominating function on
a graph if for every vertex with , there exists a vertex ,
adjacent to , such that . The weight of a Roman dominating
function is given by . The minimum weight of a Roman
dominating function on is called the Roman domination number of . In
this article we study the Roman domination number of Cartesian product graphs
and strong product graphs. More precisely, we study the relationships between
the Roman domination number of product graphs and the (Roman) domination number
of the factors
On the Roman domination number of generalized Sierpinski graphs
A map is a Roman dominating function on a
graph if for every vertex with , there exists a
vertex , adjacent to , such that . The weight of a Roman
dominating function is given by . The minimum weight
of a Roman dominating function on is called the Roman domination number of
. In this article we study the Roman domination number of Generalized
Sierpi\'{n}ski graphs . More precisely, we obtain a general upper bound
on the Roman domination number of and we discuss the tightness of this
bound. In particular, we focus on the cases in which the base graph is a
path, a cycle, a complete graph or a graph having exactly one universal vertex
On the strong metric dimension of corona product graphs and join graphs
Let be a connected graph. A vertex strongly resolves a pair ,
of vertices of if there exists some shortest path containing or
some shortest path containing . A set of vertices is a strong
resolving set for if every pair of vertices of is strongly resolved by
some vertex of . The smallest cardinality of a strong resolving set for
is called the strong metric dimension of . It is known that the problem of
computing this invariant is NP-hard. It is therefore desirable to reduce the
problem of computing the strong metric dimension of product graphs, to the
problem of computing some parameter of the factor graphs. We show that the
problem of finding the strong metric dimension of the corona product , of two graphs and , can be transformed to the problem of finding
certain clique number of . As a consequence of the study we show that if
has diameter two, then the strong metric dimension of is obtained
from the strong metric dimension of and, if is not connected or its
diameter is greater than two, then the strong metric dimension of is
obtained from the strong metric dimension of , where denotes
the trivial graph. The strong metric dimension of join graphs is also studied
On the partition dimension of unicyclic graphs
Given an ordered partition of the vertex set
of a connected graph , the \emph{partition representation} of a vertex
with respect to the partition is the vector
, where represents the
distance between the vertex and the set . A partition of is
a \emph{resolving partition} if different vertices of have different
partition representations, i.e., for every pair of vertices ,
. The \emph{partition dimension} of is the minimum
number of sets in any resolving partition for . In this paper we obtain
several tight bounds on the partition dimension of unicyclic graphs
On the complexity of computing the -metric dimension of graphs
Given a connected graph , a set is a -metric
generator for if for any two different vertices , there exist at
least vertices such that for
every . A metric generator of minimum cardinality is called a
-metric basis and its cardinality the -metric dimension of . We study
some problems regarding the complexity of some -metric dimension problems.
For instance, we show that the problem of computing the -metric dimension of
graphs is -Complete. However, the problem is solved in linear time for the
particular case of trees.Comment: 17 page
The Simultaneous Metric Dimension of Families Composed by Lexicographic Product Graphs
Let be a graph family defined on a common (labeled) vertex set
. A set is said to be a simultaneous metric generator for
if for every and every pair of different vertices
there exists such that , where
denotes the geodesic distance. A simultaneous adjacency generator for
is a simultaneous metric generator under the metric
. A minimum cardinality simultaneous metric
(adjacency) generator for is a simultaneous metric (adjacency)
basis, and its cardinality the simultaneous metric (adjacency) dimension of
. Based on the simultaneous adjacency dimension, we study the
simultaneous metric dimension of families composed by lexicographic product
graphs
On the local metric dimension of corona product graphs
A vertex is said to distinguish two vertices of a
nontrivial connected graph if the distance from to is different
from the distance from to .
A set is a local metric generator for if every two
adjacent vertices of are distinguished by some vertex in . A local
metric generator with the minimum cardinality is called a local metric basis
for and its cardinality, the local metric dimension of G. In this paper we
study the problem of finding exact values for the local metric dimension of
corona product of graphs
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