  Let $N$ be a compact, connected, nonorientable surface of genus $g$ with $n$
boundary components. Let $\lambda$ be a simplicial map of the complex of
curves, $\mathcal{C}(N)$, on $N$ which satisfies the following: $[a]$ and $[b]$
are connected by an edge in $\mathcal{C}(N)$ if and only if $\lambda([a])$ and
$\lambda([b])$ are connected by an edge in $\mathcal{C}(N)$ for every pair of
vertices $[a], [b]$ in $\mathcal{C}(N)$. We prove that $\lambda$ is induced by
a homeomorphism of $N$ if $(g, n) \in \{(1, 0), (1, 1), (2, 0)$, $(2, 1), (3,
0)\}$ or $g + n \geq 5$. Our result implies that superinjective simplicial maps
and automorphisms of $\mathcal{C}(N)$ are induced by homeomorphisms of $N$.
