A Survey on Alliances and Related Parameters in Graphs

In this paper, we show that several graph parameters are known in different areas under completely different names.More specifically, our observations connect signed domination, monopolies, $\alpha$-domination, $\alpha$-independence,positive influence domination,and a parameter associated to fast information propagationin networks to parameters related to various notions of global $r$-alliances in graphs.We also propose a new framework, called (global) $(D,O)$-alliances, not only in order to characterizevarious known variants of alliance and domination parameters, but also to suggest a unifying framework for the study of alliances and domination.Finally, we also give a survey on the mentioned graph parameters, indicating how results transfer due to our observations

Minus total domination in graphs

summary:A three-valued function $f\: V\rightarrow \{-1,0,1\}$ defined on the vertices of a graph $G=(V,E)$ is a minus total dominating function (MTDF) if the sum of its function values over any open neighborhood is at least one. That is, for every $v\in V$, $f(N(v))\ge 1$, where $N(v)$ consists of every vertex adjacent to $v$. The weight of an MTDF is $f(V)=\sum f(v)$, over all vertices $v\in V$. The minus total domination number of a graph $G$, denoted $\gamma _t^{-}(G)$, equals the minimum weight of an MTDF of $G$. In this paper, we discuss some properties of minus total domination on a graph $G$ and obtain a few lower bounds for $\gamma _t^{-}(G)$

Advances and applications in high-dimensional heuristic optimization

“Applicable to most real-world decision scenarios, multiobjective optimization is an area of multicriteria decision-making that seeks to simultaneously optimize two or more conflicting objectives. In contrast to single-objective scenarios, nontrivial multiobjective optimization problems are characterized by a set of Pareto optimal solutions wherein no solution unanimously optimizes all objectives. Evolutionary algorithms have emerged as a standard approach to determine a set of these Pareto optimal solutions, from which a decision-maker can select a vetted alternative. While easy to implement and having demonstrated great efficacy, these evolutionary approaches have been criticized for their runtime complexity when dealing with many alternatives or a high number of objectives, effectively limiting the range of scenarios to which they may be applied. This research introduces mechanisms to improve the runtime complexity of many multiobjective evolutionary algorithms, achieving state-of-the-art performance, as compared to many prominent methods from the literature. Further, the investigations here presented demonstrate the capability of multiobjective evolutionary algorithms in a complex, large-scale optimization scenario. Showcasing the approach’s ability to intelligently generate well-performing solutions to a meaningful optimization problem. These investigations advance the concept of multiobjective evolutionary algorithms by addressing a key limitation and demonstrating their efficacy in a challenging real-world scenario. Through enhanced computational efficiency and exhibited specialized application, the utility of this powerful heuristic strategy is made more robust and evident”--Abstract, page iv

Evolution and complexity: the double-edged sword

We attempt to provide a comprehensive answer to the question of whether, and when, an arrow of complexity emerges in Darwinian evolution. We note that this expression can be interpreted in different ways, including a passive, incidental growth, or a pervasive bias towards complexification. We argue at length that an arrow of complexity does indeed occur in evolution, which can be most reasonably interpreted as the result of a passive trend rather than a driven one. What, then, is the role of evolution in the creation of this trend, and under which conditions will it emerge? In the later sections of this article we point out that when certain proper conditions (which we attempt to formulate in a concise form) are met, Darwinian evolution predictably creates a sustained trend of increase in maximum complexity (that is, an arrow of complexity) that would not be possible without it; but if they are not, evolution will not only fail to produce an arrow of complexity, but may actually prevent any increase in complexity altogether. We conclude that, with regard to the growth of complexity, evolution is very much a double-edged sword

Efficient Network Domination for Life Science Applications

With the ever-increasing size of data available to researchers, traditional methods of analysis often cannot scale to match problems being studied. Often only a subset of variables may be utilized or studied further, motivating the need of techniques that can prioritize variable selection. This dissertation describes the development and application of graph theoretic techniques, particularly the notion of domination, for this purpose. In the first part of this dissertation, algorithms for vertex prioritization in the field of network controllability are studied. Here, the number of solutions to which a vertex belongs is used to classify said vertex and determine its suitability in controlling a network. Novel efficient scalable algorithms are developed and analyzed. Empirical tests demonstrate the improvement of these algorithms over those already established in the literature. The second part of this dissertation concerns the prioritization of genes for loss-of-function allele studies in mice. The International Mouse Phenotyping Consortium leads the initiative to develop a loss-of-function allele for each protein coding gene in the mouse genome. Only a small proportion of untested genes can be selected for further study. To address the need to prioritize genes, a generalizable data science strategy is developed. This strategy models genes as a gene-similarity graph, and from it selects subset that will be further characterized. Empirical tests demonstrate the method’s utility over that of pseudorandom selection and less computationally demanding methods. Finally, part three addresses the important task of preprocessing in the context of noisy public health data. Many public health databases have been developed to collect, curate, and store a variety of environmental measurements. Idiosyncrasies in these measurements, however, introduce noise to data found in these databases in several ways including missing, incorrect, outlying, and incompatible data. Beyond noisy data, multiple measurements of similar variables can introduce problems of multicollinearity. Domination is again employed in a novel graph method to handle autocorrelation. Empirical results using the Public Health Exposome dataset are reported. Together these three parts demonstrate the utility of subset selection via domination when applied to a multitude of data sources from a variety of disciplines in the life sciences

The frequency assignment problem

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis examines a wide collection of frequency assignment problems. One of the largest topics in this thesis is that of L(2,1)-labellings of outerplanar graphs. The main result in this topic is the fact that there exists a polynomial time algorithm to determine the minimum L(2,1)-span for an outerplanar graph. This result generalises the analogous result for trees, solves a stated open problem and complements the fact that the problem is NP-complete for planar graphs. We furthermore give best possible bounds on the minimum L(2,1)-span and the cyclic-L(2,1)-span in outerplanar graphs, when the maximum degree is at least eight. We also give polynomial time algorithms for solving the standard constraint matrix problem for several classes of graphs, such as chains of triangles, the wheel and a larger class of graphs containing the wheel. We furthermore introduce the concept of one-close-neighbour problems, which have some practical applications. We prove optimal results for bipartite graphs, odd cycles and complete multipartite graphs. Finally we evaluate different algorithms for the frequency assignment problem, using domination analysis. We compute bounds for the domination number of some heuristics for both the fixed spectrum version of the frequency assignment problem and the minimum span frequency assignment problem. Our results show that the standard greedy algorithm does not perform well, compared to some slightly more advanced algorithms, which is what we would expect. In this thesis we furthermore give some background and motivation for the topics being investigated, as well as mentioning several open problems.EPSR

Fast Parallel Algorithms on a Class of Graph Structures With Applications in Relational Databases and Computer Networks.

The quest for efficient parallel algorithms for graph related problems necessitates not only fast computational schemes but also requires insights into their inherent structures that lend themselves to elegant problem solving methods. Towards this objective efficient parallel algorithms on a class of hypergraphs called acyclic hypergraphs and directed hypergraphs are developed in this thesis. Acyclic hypergraphs are precisely chordal graphs and their subclasses, and they have applications in relational databases and computer networks. In this thesis, first, we present efficient parallel algorithms for the following problems on graphs. (1) determining whether a graph is strongly chordal, ptolemaic, or a block graph. If the graph is strongly chordal, determine the strongly perfect vertex elimination ordering. (2) determining the minimal set of edges needed to make an arbitrary graph strongly chordal, ptolemaic, or a block graph. (3) determining the minimum cardinality dominating set, connected dominating set, total dominating set, and the domatic number of a strongly chordal graph. Secondly, we show that the query implication problem (Q\sb1\ \to\ Q\sb2) on two queries, which is to determine whether the data retrieved by query Q\sb1 is always a subset of the data retrieved by query Q\sb2, is not even in NP and in fact complete in \Pi\sb2\sp{p}. We present several \u27fine-grain\u27 analyses of the query implication problem and show that the query implication can be solved in polynomial time given chordal queries. Thirdly, we develop efficient parallel algorithms for manipulating directed hypergraphs H such as finding a directed path in H, closure of H, and minimum equivalent hypergraph of H. We show that finding a directed path in a directed hypergraph is inherently sequential. For directed hypergraphs with fixed degree and diameter we present NC algorithms for manipulations. Directed hypergraphs are representation schemes for functional dependencies in relational databases. Finally, we also present an efficient parallel algorithm for multi-dimensional range search. We show that a set of points in a rectangular parallelepiped can be obtained in O(logn) time with only 2.log\sp2 n $-$ 10.logn + 14 processors on a EREW-PRAM. A nontrivial implementation technique on the hypercube parallel architecture is also presented. Our method can be easily generalized to the case of d-dimensional range search