  This work is the extension of the results by the author in [7] and [6] for
low-genus surfaces. Let $S$ be an orientable, connected surface of finite
topological type, with genus $g \leq 2$, empty boundary, and complexity at
least $2$; as a complement of the results of [6], we prove that any graph
endomorphism of the curve graph of $S$ is actually an automorphism. Also, as a
complement of the results in [6] we prove that under mild conditions on the
complexity of the underlying surfaces any graph morphism between curve graphs
is induced by a homeomorphism of the surfaces.
  To prove these results, we construct a finite subgraph whose union of
iterated rigid expansions is the curve graph $\mathcal{C}(S)$. The sets
constructed, and the method of rigid expansion, are closely related to
Aramayona and Leiniger's finite rigid sets in [2]. Similarly to [7], a
consequence of our proof is that Aramayona and Leininger's rigid set also
exhausts the curve graph via rigid expansions, and the combinatorial rigidity
results follow as an immediate consequence, based on the results in [6].
