For a graph G = (V,E), we consider a bipartition {V1,V2} of the vertex set V by placing constraints on the vertices as follows. For every vertex v in Vi, we place a constraint on the number of neighbors v has in Vi and a constraint on the number of neighbors it has in V3-i. Using three values, namely 0 (no neighbors are allowed), 1 (at least one neighbor is required), and X (any number of neighbors are allowed) for each of the four constraints, results in 27 distinct types of bipartitions. The goal is to characterize graphs having each of these 27 types. We give characterizations for 21 out of the 27. Three other characterizations appear in the literature. The remaining three prove to be quite difficult. For these, we develop properties and give characterization of special families.

A total dominating set in a graph G is a set S of vertices of G such that every vertex in G is adjacent to a vertex of S. We study graphs whose vertex set can be partitioned into two total dominating sets. In particular, we develop several sufficient conditions for a graph to have a vertex partition into two total dominating sets. We also show that with the exception of the cycle on five vertices, every selfcomplementary graph with minimum degree at least two has such a partition.

A total dominating set in a graph G is a set S of vertices of G such that every vertex in G is adjacent to a vertex of S. We study cubic graphs whose vertex set can be partitioned into two total dominating sets. There are infinitely many examples of connected cubic graphs that do not have such a vertex partition. In this paper, we show that the class of claw-free cubic graphs has such a partition. For an integer k at least 3, a graph is k-chordal if it does not have an induced cycle of length more than k. Chordal graphs coincide with 3-chordal graphs. We observe that for k≥6, not every graph in the class of k-chordal, connected, cubic graphs has two vertex disjoint total dominating sets. We prove that the vertex set of every 5-chordal, connected, cubic graph can be partitioned into two total dominating sets. As a consequence of this result, we observe that this property also holds for a connected, cubic graph that is chordal or 4-chordal. We also prove that cubic graphs containing a diamond as a subgraph can be partitioned into two total dominating sets.

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.