Simplicial Maps of the Complexes of Curves on Nonorientable Surfaces

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 $N$.