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.