86 research outputs found
Rainbow domination and related problems on some classes of perfect graphs
Let and let be a graph. A function is a rainbow function if, for every vertex with
, . The rainbow domination number
is the minimum of over all rainbow
functions. We investigate the rainbow domination problem for some classes of
perfect graphs
The Upper Domatic Number of a Graph
Let (Formula presented.) be a graph. For two disjoint sets of vertices (Formula presented.) and (Formula presented.), set (Formula presented.) dominates set (Formula presented.) if every vertex in (Formula presented.) is adjacent to at least one vertex in (Formula presented.). In this paper we introduce the upper domatic number (Formula presented.), which equals the maximum order (Formula presented.) of a vertex partition (Formula presented.) such that for every (Formula presented.), (Formula presented.), either (Formula presented.) dominates (Formula presented.) or (Formula presented.) dominates (Formula presented.), or both. We study properties of the upper domatic number of a graph, determine bounds on (Formula presented.), and compare (Formula presented.) to a related parameter, the transitivity (Formula presented.) of (Formula presented.)
Disjoint Total Dominating Sets in Near-Triangulations
We show that every simple planar near-triangulation with minimum degree at
least three contains two disjoint total dominating sets. The class includes all
simple planar triangulations other than the triangle. This affirms a conjecture
of Goddard and Henning [Thoroughly dispersed colorings, J. Graph Theory, 88
(2018) 174-191]
Determining Distributions of Security Means for WSNs based on the Model of a Neighbourhood Watch
Neighbourhood watch is a concept that allows a community to distribute a
complex security task in between all members. Members of the community carry
out individual security tasks to contribute to the overall security of it. It
reduces the workload of a particular individual while securing all members and
allowing them to carry out a multitude of security tasks. Wireless sensor
networks (WSNs) are composed of resource-constraint independent battery driven
computers as nodes communicating wirelessly. Security in WSNs is essential.
Without sufficient security, an attacker is able to eavesdrop the
communication, tamper monitoring results or deny critical nodes providing their
service in a way to cut off larger network parts. The resource-constraint
nature of sensor nodes prevents them from running full-fledged security
protocols. Instead, it is necessary to assess the most significant security
threats and implement specialised protocols. A neighbourhood-watch inspired
distributed security scheme for WSNs has been introduced by Langend\"orfer. Its
goal is to increase the variety of attacks a WSN can fend off. A framework of
such complexity has to be designed in multiple steps. Here, we introduce an
approach to determine distributions of security means on large-scale static
homogeneous WSNs. Therefore, we model WSNs as undirected graphs in which two
nodes connected iff they are in transmission range. The framework aims to
partition the graph into distinct security means resulting in the targeted
distribution. The underlying problems turn out to be NP hard and we attempt to
solve them using linear programs (LPs). To evaluate the computability of the
LPs, we generate large numbers of random {\lambda}-precision unit disk graphs
(UDGs) as representation of WSNs. For this purpose, we introduce a novel
{\lambda}-precision UDG generator to model WSNs with a minimal distance in
between nodes
A Study Of The Upper Domatic Number Of A Graph
Given a graph G we can partition the vertices of G in to k disjoint sets. We say a set A of vertices dominates another set of vertices, B, if for every vertex in B there is some adjacent vertex in A. The upper domatic number of a graph G is written as D(G) and defined as the maximum integer k such that G can be partitioned into k sets where for every pair of sets A and B either A dominates B or B dominates A or both. In this thesis we introduce the upper domatic number of a graph and provide various results on the properties of the upper domatic number, notably that D(G) is less than or equal to the maximum degree of G, as well as relating it to other well-studied graph properties such as the achromatic, pseudoachromatic, and transitive numbers
Signed star k-domatic number of a graph
Let be a simple graph without isolated vertices with vertex set
and edge set and let be a positive integer. A function is said to be a signed star -dominating function on if
for every vertex of , where
. A set of
signed star -dominating functions on with the property that
for each , is called a signed
star -dominating family (of functions) on . The maximum number of
functions in a signed star -dominating family on is the signed
star -domatic number of , denoted by
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