Minimum Degree up to Local Complementation: Bounds, Parameterized Complexity, and Exact Algorithms

The local minimum degree of a graph is the minimum degree that can be reached by means of local complementation. For any n, there exist graphs of order n which have a local minimum degree at least 0.189n, or at least 0.110n when restricted to bipartite graphs. Regarding the upper bound, we show that for any graph of order n, its local minimum degree is at most 3n/8+o(n) and n/4+o(n) for bipartite graphs, improving the known n/2 upper bound. We also prove that the local minimum degree is smaller than half of the vertex cover number (up to a logarithmic term). The local minimum degree problem is NP-Complete and hard to approximate. We show that this problem, even when restricted to bipartite graphs, is in W[2] and FPT-equivalent to the EvenSet problem, which W[1]-hardness is a long standing open question. Finally, we show that the local minimum degree is computed by a O*(1.938^n)-algorithm, and a O*(1.466^n)-algorithm for the bipartite graphs

Ideal homogeneous access structures constructed from graphs

Starting from a new relation between graphs and secret sharing schemes introduced by Xiao, Liu and Zhang, we show a method to construct more general ideal homogeneous access structures. The method has some advantages: it efficiently gives an ideal homogeneous access structure for the desired rank, and some conditions can be imposed (such as forbidden or necessary subsets of players), even if the exact composition of the resulting access structure cannot be fully controlled. The number of homogeneous access structures that can be constructed in this way is quite limited; for example, we show that (t, l)-threshold access structures can be constructed from a graph only when t = 1, t = l - 1 or t = l.Peer ReviewedPostprint (published version

Secure message transmission in the general adversary model

The problem of secure message transmission (SMT), due to its importance in both practice and theory, has been studied extensively. Given a communication network in which a sender S and a receiver R are indirectly connected by unreliable and distrusted channels, the aim of SMT is to enable messages to be transmitted from S to R with a reasonably high level of privacy and reliability. SMT must be achieved in the presence of a Byzantine adversary who has unlimited computational power and can corrupt the transmission. In the general adversary model, the adversary is characterized by an adversary structure. We study two diff�erent measures of security: perfect (PSMT) and almost perfect (APSMT). Moreover, reliable (but not private) message transmission (RMT) are considered as a specifi�c part of SMT. In this thesis, we study RMT, APSMT and PSMT in two di�fferent network settings: point-to-point and multicast. To prepare the study of SMT in these two network settings, we present some ideas and observations on secret sharing schemes (SSSs), generalized linear codes and critical paths. First, we prove that the error-correcting capability of an almost perfect SSS is the same as a perfect SSS. Next, we regard general access structures as linear codes, and introduce some new properties that allow us to construct pseudo-basis for efficient PSMT protocol design. In addition, we de�fine adversary structures over "critical paths", and observe their properties. Having these new developments, the contributions on SMT in the aforementioned two network settings can be presented as follows. The results on SMT in point-to-point networks are obtained in three aspects. First, we show a Guessing Attack on some existing PSMT protocols. This attack is critically important to the design of PSMT protocols in asymmetric networks. Second, we determine necessary and sufficient conditions for di�fferent levels of RMT and APSMT. In particular, by applying the result on almost perfect SSS, we show that relaxing the requirement of privacy does not weaken the minimal network connectivity. Our �final contribution in the point-to-point model is to give the �first ever efficient, constant round PSMT protocols in the general adversary model. These protocols are designed using linear codes and critical paths, and they signifi�cantly improve some previous results in terms of communication complexity and round complexity. Regarding SMT in multicast networks, we solve a problem that has been open for over a decade. That is, we show the necessary and sufficient conditions for all levels of SMT in di�fferent adversary models. First, we give an Extended Characterization of the network graphs based on our observation on the eavesdropping and separating activities of the adversary. Next, we determine the necessary and sufficient conditions for SMT in the general adversary model with the new Extended Characterization. Finally, we apply the results to the threshold adversary model to completely solve the problem of SMT in general multicast network graphs

Applications of graph theory to wireless networks and opinion analysis

La teoría de grafos es una rama importante dentro de la matemática discreta. Su uso ha aumentado recientemente dada la conveniencia de los grafos para estructurar datos, para analizarlos y para generarlos a través de modelos. El objetivo de esta tesis es aplicar teoría de grafos a la optimización de redes inalámbricas y al análisis de opinión. El primer conjunto de contribuciones de esta tesis versa sobre la aplicación de teoría de grafos a redes inalámbricas. El rendimiento de estas redes depende de la correcta distribución de canales de frecuencia en un espacio compartido. Para optimizar estas redes se proponen diferentes técnicas, desde la aplicación de heurísticas como simulated annealing a la negociación automática. Cualquiera de estas técnicas requiere un modelo teórico de la red inalámbrica en cuestión. Nuestro modelo de redes Wi-Fi utiliza grafos geométricos para este propósito. Los vértices representan los dispositivos de la red, sean clientes o puntos de acceso, mientras que las aristas representan las señales entre dichos dispositivos. Estos grafos son de tipo geométrico, por lo que los vértices tienen posición en el espacio, y las aristas tienen longitud. Con esta estructura y la aplicación de un modelo de propagación y de uso, podemos simular redes inalámbricas y contribuir a su optimización. Usando dicho modelo basado en grafos, hemos estudiado el efecto de la interferencia cocanal en redes Wi-Fi 4 y mostramos una mejora de rendimiento asociada a la técnica de channel bonding cuando se usa en regiones donde hay por lo menos 13 canales disponibles. Por otra parte, en esta tesis doctoral hemos aplicado teoría de grafos al análisis de opinión dentro de la línea de investigación de SensoGraph, un método con el que se realiza un análisis de opinión sobre un conjunto de elementos usando grafos de proximidad, lo que permite manejar grandes conjuntos de datos. Además, hemos desarrollado un método de análisis de opinión que emplea la asignación manual de aristas y distancias en un grafo para estudiar la similaridad entre las muestras dos a dos. Adicionalmente, se han explorado otros temas sin relación con los grafos, pero que entran dentro de la aplicación de las matemáticas a un problema de la ingeniería telemática. Se ha desarrollado un sistema de votación electrónica basado en mixnets, secreto compartido de Shamir y cuerpos finitos. Dicha propuesta ofrece un sistema de verificación numérico novedoso a la vez que mantiene las propiedades esenciales de los sistemas de votación