  A relational structure is (connected-)homogeneous if every isomorphism
between finite (connected) substructures extends to an automorphism of the
structure. We investigate notions which generalise (connected-)homogeneity,
where "isomorphism" may be replaced by "homomorphism" or "monomorphism" in the
definition. Specifically, we study the classes of finite
connected-homomorphism-homogeneous graphs, with the aim of producing
classifications. The main result is a classification of the finite C-HH graphs,
where a graph G is C-HH if every homomorphism from a finite connected induced
subgraph of G into G extends to an endomorphism of G. The finite C-II
(connected-homogeneous) graphs were classified by Gardiner in 1976, and from
this we obtain classifications of the finite C-HI and C-MI finite graphs.
Although not all the classes of finite connected-homomorphism-homogeneous
graphs are completely characterised, we may still obtain the final hierarchy
picture for these classes.
