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Abstract

Homogeneous structures are a well studied research area and have variety uses like constructions in model theory and permutation group theory. Recently Cameron and Nesetril have introduced homomorphism homogeneity by incorporating homomorphisms in the definition of homogeneity. This has attracted a fair bit of attention from the research community and a growing amount of research has been done in this area for different relational structures. The first goal of this thesis is to investigate the different classes of homomorphism homogeneous simple undirected graphs with respect to different kinds of homomorphisms and study the relations between these classes. Although homogeneous graphs are heavily analyzed, little has been done for homomorphism homogeneous graphs. Cameron and Nesetril posed two open questions when they first defined these graphs. We answer both questions and also attempt to classify the homomorphism homogeneous graphs. This, we believe, opens up future possibilities for more analysis of these structures. In the thesis we also treat the category of graphs with loop allowed and further extend the idea of homogeneity by expanding the list of homomorphisms that are taken into consideration in the definitions.