  Let $R$ be a compact, connected, orientable surface of genus $g$ with $n$
boundary components with $g \geq 2$, $n \geq 0$. Let $\mathcal{N}(R)$ be the
nonseparating curve graph, $\mathcal{C}(R)$ be the curve graph and
$\mathcal{HT}(R)$ be the Hatcher-Thurston graph of $R$. We prove that if
$\lambda : \mathcal{N}(R) \rightarrow\mathcal{N}(R)$ is an edge-preserving map,
then $\lambda$ is induced by a homeomorphism of $R$. We prove that if $\theta :
\mathcal{C}(R) \rightarrow \mathcal{C}(R)$ is an edge-preserving map, then
$\theta$ is induced by a homeomorphism of $R$. We prove that if $R$ is closed
and $\tau: \mathcal{HT}(R) \rightarrow\mathcal{HT}(R)$ is a rectangle
preserving map, then $\tau$ is induced by a homeomorphism of $R$. We also prove
that these homeomorphisms are unique up to isotopy when $(g, n) \neq (2, 0)$.
