The poset of morphism-extension classes of countable graphs

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.