Any reasonably large group of individuals, families, states, and parties exhibits the phenomenon of subgroup formations within the group such that the members of each group have a strong connection or bonding between each other. The reasons of the formation of these subgroups that we call alliances differ in different situations, such as, kinship and friendship (in the case of individuals), common economic interests (for both individuals and states), common political interests, and geographical proximity. This structure of alliances is not only prevalent in social networks, but it is also an important characteristic of similarity networks of natural and unnatural objects. (A similarity network defines the links between two objects based on their similarities). Discovery of such structure in a data set is called clustering or unsupervised learning and the ability to do it automatically is desirable for many applications in the areas of pattern recognition, computer vision, artificial intelligence, behavioral and social sciences, life sciences, earth sciences, medicine, and information theory. In this dissertation, we study a graph theoretical model of alliances where an alliance of the vertices of a graph is a set of vertices in the graph, such that every vertex in the set is adjacent to equal or more vertices inside the set than the vertices outside it. We study the problem of partitioning a graph into alliances and identify classes of graphs that have such a partition. We present results on the relationship between the existence of such a partition and other well known graph parameters, such as connectivity, subgraph structure, and degrees of vertices. We also present results on the computational complexity of finding such a partition. An alliance cover set is a set of vertices in a graph that contains at least one vertex from every alliance of the graph. The complement of an alliance cover set is an alliance free set, that is, a set that does not contain any alliance as a subset. We study the properties of these sets and present tight bounds on their cardinalities. In addition, we also characterize the graphs that can be partitioned into alliance free and alliance cover sets. Finally, we present an approximate algorithm to discover alliances in a given graph. At each step, the algorithm finds a partition of the vertices into two alliances such that the alliances are strongest among all such partitions. The strength of an alliance is defined as a real number p, such that every vertex in the alliance has at least p times more neighbors in the set than its total number of neighbors in the graph). We evaluate the performance of the proposed algorithm on standard data sets.

Submitted by Jaqueline Silva (jtas29@gmail.com) on 2014-09-04T17:02:47Z No. of bitstreams: 2 Dissertacao Elisangela Silva Dias.pdf: 846122 bytes, checksum: 357f425f14050b1601ed04cbcd4d9165 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2014-09-04T17:02:47Z (GMT). No. of bitstreams: 2 Dissertacao Elisangela Silva Dias.pdf: 846122 bytes, checksum: 357f425f14050b1601ed04cbcd4d9165 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2010-03-26 / A defensive alliance in graph G = (V;E) is a set of vertices S V satisfying the condition that every vertex v 2 S has at most one more neighbor in V 􀀀S than S. Due to this type of alliance, the vertices in S together defend themselves to the vertices in V 􀀀S. This dissertation introduces the basic concepts for the understanding of alliances in graphs, along with a variety of alliances and their numbers and provides some mathematical properties for these alliances, focusing mainly on defensive alliances in graphs. It shows theorems, corollaries, lemmas, propositions and observations with appropriate proofs with respect to the minimum degree of a graph G d(G), the maximum degree D(G), the algebraic connectivity μ, the total dominanting set gt(G), the eccentricity, the edge connectivity l(G), the chromatic number c(G), the (vertex) independence number b0(G), the vertex connectivity k(G), the order of the largest clique w(G) and the domination number g(G). It also shows a generalization of defensive alliances, called defensive kalliance, and the definition and properties of a security set in G. A secure set S V of graph G = (V;E) is a set whose every nonempty subset can be successfully defended of an attack, under appropriate definitions of “attack” and “defence”. / Uma aliança defensiva no grafo G = (V;E) é um conjunto de vértices S V satisfazendo a condição de que todo vértice v 2 S tem no máximo um vizinho a mais em V 􀀀S que em S. Devido a este tipo de aliança, os vértices em S juntam para se defenderem dos vértices em V 􀀀S. Nesta dissertação, são introduzidos os conceitos básicos para o entendimentos das alianças em grafos, junto com uma variedade de tipos de alianças e seus respectivos números, bem como são fornecidas algumas propriedades matemáticas para estas alianças, focando principalmente nas alianças defensivas em grafos. Apresentamos teoremas, corolários, lemas, proposições e observações com as devidas provas com relação ao grau mínimo de um grafo G d(G), ao grau máximo D(G), à conectividade algébrica μ, ao conjunto dominante total gt(G), à excentricidade, à conectividade de arestas l(G), ao número cromático c(G), ao número de independência (de vértices) b0(G), à conectividade de vértices k(G), à ordem da maior clique w(G) e ao número de dominação g(G). Também é mostrada a generalização de alianças defensivas, chamada k-aliança defensiva, e a definição e propriedades de um conjunto seguro em G. Um conjunto seguro S V do grafo G = (V;E) é um conjunto no qual todo subconjunto não-vazio pode ser defendido com sucesso de um ataque, sob as definições apropriadas de “ataque” e “defesa”.