1,236 research outputs found
Zero forcing in iterated line digraphs
Zero forcing is a propagation process on a graph, or digraph, defined in
linear algebra to provide a bound for the minimum rank problem. Independently,
zero forcing was introduced in physics, computer science and network science,
areas where line digraphs are frequently used as models. Zero forcing is also
related to power domination, a propagation process that models the monitoring
of electrical power networks.
In this paper we study zero forcing in iterated line digraphs and provide a
relationship between zero forcing and power domination in line digraphs. In
particular, for regular iterated line digraphs we determine the minimum
rank/maximum nullity, zero forcing number and power domination number, and
provide constructions to attain them. We conclude that regular iterated line
digraphs present optimal minimum rank/maximum nullity, zero forcing number and
power domination number, and apply our results to determine those parameters on
some families of digraphs often used in applications
Orientable domination in product-like graphs
The orientable domination number, , of a graph is the
largest domination number over all orientations of . In this paper, is studied on different product graphs and related graph operations. The
orientable domination number of arbitrary corona products is determined, while
sharp lower and upper bounds are proved for Cartesian and lexicographic
products. A result of Chartrand et al. from 1996 is extended by establishing
the values of for arbitrary positive integers
and . While considering the orientable domination number of
lexicographic product graphs, we answer in the negative a question concerning
domination and packing numbers in acyclic digraphs posed in [Domination in
digraphs and their direct and Cartesian products, J. Graph Theory 99 (2022)
359-377]
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
Eternal dominating sets on digraphs and orientations of graphs
We study the eternal dominating number and the m-eternal dominating number on
digraphs. We generalize known results on graphs to digraphs. We also consider
the problem "oriented (m-)eternal domination", consisting in finding an
orientation of a graph that minimizes its eternal dominating number. We prove
that computing the oriented eternal dominating number is NP-hard and
characterize the graphs for which the oriented m-eternal dominating number is
2. We also study these two parameters on trees, cycles, complete graphs,
complete bipartite graphs, trivially perfect graphs and different kinds of
grids and products of graphs.Comment: 34 page
The Distribution of the Domination Number of a Family of Random Interval Catch Digraphs
We study a new kind of proximity graphs called proportional-edge proximity
catch digraphs (PCDs)in a randomized setting. PCDs are a special kind of random
catch digraphs that have been developed recently and have applications in
statistical pattern classification and spatial point pattern analysis. PCDs are
also a special type of intersection digraphs; and for one-dimensional data, the
proportional-edge PCD family is also a family of random interval catch
digraphs. We present the exact (and asymptotic) distribution of the domination
number of this PCD family for uniform (and non-uniform) data in one dimension.
We also provide several extensions of this random catch digraph by relaxing the
expansion and centrality parameters, thereby determine the parameters for which
the asymptotic distribution is non-degenerate. We observe sudden jumps (from
degeneracy to non-degeneracy or from a non-degenerate distribution to another)
in the asymptotic distribution of the domination number at certain parameter
values.Comment: 29 pages, 3 figure
Domination and location in twin-free digraphs
A dominating set in a digraph is a set of vertices such that every vertex
is either in or has an in-neighbour in . A dominating set of a
digraph is locating-dominating if every vertex not in has a unique set of
in-neighbours within . The location-domination number of a
digraph is the smallest size of a locating-dominating set of . We
investigate upper bounds on in terms of the order of . We
characterize those digraphs with location-domination number equal to the order
or the order minus one. Such digraphs always have many twins: vertices with the
same (open or closed) in-neighbourhoods. Thus, we investigate the value of
in the absence of twins and give a general method for
constructing small locating-dominating sets by the means of special dominating
sets. In this way, we show that for every twin-free digraph of order ,
holds, and there exist twin-free digraphs
with . If moreover is a tournament or is
acyclic, the bound is improved to ,
which is tight in both cases
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