Rainbow domination and related problems on some classes of perfect graphs

Let $k \in \mathbb{N}$ and let $G$ be a graph. A function $f: V(G) \rightarrow 2^{[k]}$ is a rainbow function if, for every vertex $x$ with $f(x)=\emptyset$, $f(N(x)) =[k]$. The rainbow domination number $\gamma_{kr}(G)$ is the minimum of $\sum_{x \in V(G)} |f(x)|$ 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]

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

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 $n$ 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

Families with infants: a general approach to solve hard partition problems

We introduce a general approach for solving partition problems where the goal is to represent a given set as a union (either disjoint or not) of subsets satisfying certain properties. Many NP-hard problems can be naturally stated as such partition problems. We show that if one can find a large enough system of so-called families with infants for a given problem, then this problem can be solved faster than by a straightforward algorithm. We use this approach to improve known bounds for several NP-hard problems as well as to simplify the proofs of several known results. For the chromatic number problem we present an algorithm with $O^*((2-\varepsilon(d))^n)$ time and exponential space for graphs of average degree $d$. This improves the algorithm by Bj\"{o}rklund et al. [Theory Comput. Syst. 2010] that works for graphs of bounded maximum (as opposed to average) degree and closes an open problem stated by Cygan and Pilipczuk [ICALP 2013]. For the traveling salesman problem we give an algorithm working in $O^*((2-\varepsilon(d))^n)$ time and polynomial space for graphs of average degree $d$. The previously known results of this kind is a polyspace algorithm by Bj\"{o}rklund et al. [ICALP 2008] for graphs of bounded maximum degree and an exponential space algorithm for bounded average degree by Cygan and Pilipczuk [ICALP 2013]. For counting perfect matching in graphs of average degree~$d$ we present an algorithm with running time $O^*((2-\varepsilon(d))^{n/2})$ and polynomial space. Recent algorithms of this kind due to Cygan, Pilipczuk [ICALP 2013] and Izumi, Wadayama [FOCS 2012] (for bipartite graphs only) use exponential space.Comment: 18 pages, a revised version of this paper is available at http://arxiv.org/abs/1410.220