Partitioning a graph into disjoint cliques and a triangle-free graph

A graph G=(V,E) is partitionable if there exists a partition {A,B} of V such that A induces a disjoint union of cliques (i.e., G[A] is P_3-free) and B induces a triangle-free graph (i.e., G[B] is K_3-free). In this paper we investigate the computational complexity of deciding whether a graph is partitionable. The problem is known to be NP-complete on arbitrary graphs. Here it is proved that if a graph G is bull-free, planar, perfect, K_4-free or does not contain certain holes then deciding whether G is partitionable is NP-complete. This answers an open question posed by Thomassé, Trotignon and Vušković. In contrast a finite list of forbidden induced subgraphs is given for partitionable cographs

Grundy number on P4-classes

International audienceIn this article, we define a new class of graphs, the fat-extended P4 -laden graphs, and we show a polynomial time algorithm to determine the Grundy number of the graphs in this class. This result implies that the Grundy number can be found in polynomial time for any graph of the following classes: P4 -reducible, extended P4 -reducible, P4 -sparse, extended P4 -sparse, P4 -extendible, P4 -lite, P4 -tidy, P4 -laden and extended P4 -laden, which are all strictly contained in the fat-extended P4 - laden class

On the Grundy number of graphs with few P4's

International audienceThe Grundy number of a graph G is the largest number of colors used by any execution of the greedy algorithm to color G. The problem of determining the Grundy number of G is polynomial if G is a P4-free graph and NP-hard if G is a P5-free graph. In this article, we define a new class of graphs, the fat-extended P4-laden graphs, and we show a polynomial time algorithm to determine the Grundy number of any graph in this class. Our class intersects the class of P5-free graphs and strictly contains the class of P4-free graphs. More precisely, our result implies that the Grundy number can be computed in polynomial time for any graph of the following classes: P4-reducible, extended P4-reducible, P4-sparse, extended P4-sparse, P4-extendible, P4-lite, P4-tidy, P4-laden and extended P4-laden, which are all strictly contained in the fat-extended P4-laden class

Topics in graph colouring and extremal graph theory

In this thesis we consider three problems related to colourings of graphs and one problem in extremal graph theory. Let $G$ be a connected graph with $n$ vertices and maximum degree $\Delta(G)$. Let $R_k(G)$ denote the graph with vertex set all proper $k$-colourings of $G$ and two $k$-colourings are joined by an edge if they differ on the colour of exactly one vertex. Our first main result states that $R_{\Delta(G)+1}(G)$ has a unique non-trivial component with diameter $O(n^2)$. This result can be viewed as a reconfigurations analogue of Brooks' Theorem and completes the study of reconfigurations of colourings of graphs with bounded maximum degree. A Kempe change is the operation of swapping some colours $a$, $b$ of a component of the subgraph induced by vertices with colour $a$ or $b$. Two colourings are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. Our second main result states that all $\Delta(G)$-colourings of a graph $G$ are Kempe equivalent unless $G$ is the complete graph or the triangular prism. This settles a conjecture of Mohar (2007). Motivated by finding an algorithmic version of a structure theorem for bull-free graphs due to Chudnovsky (2012), we consider the computational complexity of deciding if the vertices of a graph can be partitioned into two parts such that one part is triangle-free and the other part is a collection of complete graphs. We show that this problem is NP-complete when restricted to five classes of graphs (including bull-free graphs) while polynomial-time solvable for the class of cographs. Finally we consider a graph-theoretic version formulated by Holroyd, Spencer and Talbot (2007) of the famous Erd\H{o}s-Ko-Rado Theorem in extremal combinatorics and obtain some results for the class of trees

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor

The Stylometric Processing of Sensory Open Source Data

This research project’s end goal is on the Lone Wolf Terrorist. The project uses an exploratory approach to the self-radicalisation problem by creating a stylistic fingerprint of a person's personality, or self, from subtle characteristics hidden in a person's writing style. It separates the identity of one person from another based on their writing style. It also separates the writings of suicide attackers from ‘normal' bloggers by critical slowing down; a dynamical property used to develop early warning signs of tipping points. It identifies changes in a person's moods, or shifts from one state to another, that might indicate a tipping point for self-radicalisation. Research into authorship identity using personality is a relatively new area in the field of neurolinguistics. There are very few methods that model how an individual's cognitive functions present themselves in writing. Here, we develop a novel algorithm, RPAS, which draws on cognitive functions such as aging, sensory processing, abstract or concrete thinking through referential activity emotional experiences, and a person's internal gender for identity. We use well-known techniques such as Principal Component Analysis, Linear Discriminant Analysis, and the Vector Space Method to cluster multiple anonymous-authored works. Here we use a new approach, using seriation with noise to separate subtle features in individuals. We conduct time series analysis using modified variants of 1-lag autocorrelation and the coefficient of skewness, two statistical metrics that change near a tipping point, to track serious life events in an individual through cognitive linguistic markers. In our journey of discovery, we uncover secrets about the Elizabethan playwrights hidden for over 400 years. We uncover markers for depression and anxiety in modern-day writers and identify linguistic cues for Alzheimer's disease much earlier than other studies using sensory processing. In using these techniques on the Lone Wolf, we can separate their writing style used before their attacks that differs from other writing

The structure and dynamics of multilayer networks

In the past years, network theory has successfully characterized the interaction among the constituents of a variety of complex systems, ranging from biological to technological, and social systems. However, up until recently, attention was almost exclusively given to networks in which all components were treated on equivalent footing, while neglecting all the extra information about the temporal- or context-related properties of the interactions under study. Only in the last years, taking advantage of the enhanced resolution in real data sets, network scientists have directed their interest to the multiplex character of real-world systems, and explicitly considered the time-varying and multilayer nature of networks. We offer here a comprehensive review on both structural and dynamical organization of graphs made of diverse relationships (layers) between its constituents, and cover several relevant issues, from a full redefinition of the basic structural measures, to understanding how the multilayer nature of the network affects processes and dynamics.Comment: In Press, Accepted Manuscript, Physics Reports 201

Command and Control in the Information Age: A Case Study of a Representative Air Power Command and Control Node

As operations command structures change, it is important to be able to explore and understand their fundamental nature; researchers should unearth the gestalt nature of the operational node. The organizational structure and the infrastructure can significantly affect overall command and control (C2) performance. Thus, it is necessary to develop understanding of effectiveness of the technical network and the people using the system as a whole. The purpose of this research is to conduct an analysis of a representative Air Power Operational C2 node, create and use a repeatable method, and present the results as a case study to elicit fundamental understanding. I posit that there is a recognizable (and discoverable) relationship between the social (human) network and technical supporting network. Examining the system under change can result in an understanding of this relationship. In this work, I enhanced an existing simulation tool to investigate the effects of organizational structure on task effectiveness. The primary research question examined is how a representative AOC system changes varying noise and system fragmentation when operating in two different organizational constructs. Network-Enabled Capability (as the term is used in NATO), Network Centric Operations, or Edge Organizations, is a core C2 transformation predicated upon a set of network-centric tenets. These tenets form the intellectual foundation for ongoing transformations. The secondary research question is to determine if these tenets are unbound, and what elucidation results if they are not. This research produces four significant contributions to Operational Command and Control and Engineering Management disciplines. First, I combined social networking theory and information theory into a single lens for evaluation. By using this new concept, I will be able to accomplish a quantitative evaluation by something other than mission treads, field exercise, historical evaluation, or actual combat. Second, I used both information theory and social networking concepts in a non-traditional setting. Third, I hope this research will start the process required to gain the knowledge to achieve some sort of future C2 structure. Fourth, this research suggests directions for future research to enhance understanding of core Operational Command and Control concepts