  Let $\mathrm{XY_{L,T}}$ denote the class of countably infinite $L$-structures
that satisfy the axioms $T$ and in which all homomorphisms of type X (these
could be homomorphisms, monomorphisms, or isomorphisms) between finite
substructures of $M$ are restrictions of an endomorphism of $M$ of type Y (for
example, an automorphism or a surjective endomorphism). Lockett and Truss
introduced 18 such morphism-extension classes for relational structures. For a
given pair $L,T$, however, two or more morphism-extension properties may define
the same class of structures.
  In this paper, we establish all equalities and inequalities between
morphism-extension classes of countable (undirected, loopless) graphs.
