52 research outputs found
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
Outer Independent Double Italian Domination of Some Graph Products
An outer independent double Italian dominating function on a graph is a function for which each vertex with then and vertices assigned under are independent. The outer independent double Italian domination number is the minimum weight of an outer independent double Italian dominating function of graph . In this work, we present some contributions to the study of outer independent double Italian domination of three graph products. We characterize the Cartesian product, lexicographic product and direct product of custom graphs in terms of this parameter. We also provide the best possible upper and lower bounds for these three products for arbitrary graphs
On the diameter of dot-critical graphs
A graph G is -dot-critical (totaly -dot-critical) if is dot-critical (totaly dot-critical) and the domination number is . In the paper [T. Burtona, D. P. Sumner, Domination dot-critical graphs, Discrete Math, 306 (2006), 11-18] the following question is posed: What are the best bounds for the diameter of a -dot-critical graph and a totally -dot-critical graph with no critical vertices for ? We find the best bound for the diameter of a -dot-critical graph, where and we give a family of -dot-critical graphs (with no critical vertices) with sharp diameter for even
Double domination and total -domination in digraphs and their dual problems
A subset of vertices of a digraph is a double dominating set (total
-dominating set) if every vertex not in is adjacent from at least two
vertices in , and every vertex in is adjacent from at least one vertex
in (the subdigraph induced by has no isolated vertices). The double
domination number (total -domination number) of a digraph is the minimum
cardinality of a double dominating set (total -dominating set) in . In
this work, we investigate these concepts which can be considered as two
extensions of double domination in graphs to digraphs, along with the concepts
-limited packing and total -limited packing which have close
relationships with the above-mentioned concepts
Upper bounds for covering total double Roman domination
Let G = (V, E) be a finite simple graph where V = V (G) and E = E(G). Suppose that G has no isolated vertex. A covering total double Roman dominating function (CT DRD function) f of G is a total double Roman dominating function (T DRD function) of G for which the set {v ∈ V (G)|f(v) ≠0} is a covering set. The covering total double Roman domination number γctdR(G) is the minimum weight of a CT DRD function on G. In this work, we present some contributions to the study of γctdR(G)-function of graphs. For the non star trees T, we show that γctdR(T) ≤ 4n(T )+5s(T )−4l(T )/3, where n(T), s(T) and l(T) are the order, the number of support vertices and the number of leaves of T respectively. Moreover, we characterize trees T achieve this bound. Then we study the upper bound of the 2-edge connected graphs and show that, for a 2-edge connected graphs G, γctdR(G) ≤ 4n/3 and finally, we show that, for a simple graph G of order n with δ(G) ≥ 2, γctdR(G) ≤ 4n/3 and this bound is sharp.Publisher's Versio
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