35 research outputs found
The Signed Roman Domatic Number of a Digraph
Let be a finite and simple digraph with vertex set .A {\em signed Roman dominating function} on the digraph isa function such that for every , where consists of andall inner neighbors of , and every vertex for which has an innerneighbor for which . A set of distinct signedRoman dominating functions on with the property that for each, is called a {\em signed Roman dominating family} (of functions) on . The maximumnumber of functions in a signed Roman dominating family on is the {\em signed Roman domaticnumber} of , denoted by . In this paper we initiate the study of signed Romandomatic number in digraphs and we present some sharp bounds for . In addition, wedetermine the signed Roman domatic number of some digraphs. Some of our results are extensionsof well-known properties of the signed Roman domatic number of graphs
Signed total double Roman dominatıon numbers in digraphs
Let D = (V, A) be a finite simple digraph. A signed total double Roman dominating function (STDRD-function) on the digraph D is a function f : V (D) → {−1, 1, 2, 3} satisfying the following conditions: (i) P x∈N−(v) f(x) ≥ 1 for each v ∈ V (D), where N−(v) consist of all in-neighbors of v, and (ii) if f(v) = −1, then the vertex v must have at least two in-neighbors assigned 2 under f or one in-neighbor assigned 3 under f, while if f(v) = 1, then the vertex v must have at least one in-neighbor assigned 2 or 3 under f. The weight of a STDRD-function f is the value P x∈V (D) f(x). The signed total double Roman domination number (STDRD-number) γtsdR(D) of a digraph D is the minimum weight of a STDRD-function on D. In this paper we study the STDRD-number of digraphs, and we present lower and upper bounds for γtsdR(D) in terms of the order, maximum degree and chromatic number of a digraph. In addition, we determine the STDRD-number of some classes of digraphs.Publisher's Versio
Limited packings: related vertex partitions and duality issues
A -limited packing partition (LP partition) of a graph is a
partition of into -limited packing sets. We consider the LP
partitions with minimum cardinality (with emphasis on ). The minimum
cardinality is called LP partition number of and denoted by
. This problem is the dual problem of -tuple domatic
partitioning as well as a generalization of the well-studied -distance
coloring problem in graphs.
We give the exact value of for trees and bound it for
general graphs. A section of this paper is devoted to the dual of this problem,
where we give a solution to an open problem posed in . We also revisit
the total limited packing number in this paper and prove that the problem of
computing this parameter is NP-hard even for some special families of graphs.
We give some inequalities concerning this parameter and discuss the difference
between TLP number and LP number with emphasis on trees
Characterizing Heterogeneity in Cooperative Networks From a Resource Distribution View-Point
© by International Press. First published in Communications in Information and Systems, Vol. 14, no. 1, 2014, by International Press.DOI: http://dx.doi.org/10.4310/CIS.2014.v14.n1.a1A network of agents in which agents with a diverse set of resources
or capabilities interact and coordinate with each other to accomplish various tasks constitutes a heterogeneous cooperative network. In this paper, we investigate heterogeneity in terms of resources allocated to agents within the network. The objective is to distribute resources in such a way that every agent in the network
should be able to utilize all these resources through local interactions. In particular, we formulate a graph coloring problem in which
each node is assigned a subset of labels from a labeling set, and a graph is considered to be completely heterogeneous whenever all the labels in the labeling set are available in the closed neighborhood of every node. The total number of different resources that can be accommodated within a system under this setting depends
on the underlying graph structure of the network. This paper provides an analysis of the assignment of multiple resources to nodes and the effect of these assignments on the overall heterogeneity of the network