We prove that each injective simplicial map from the complex of arcs of a compact, connected, nonorientable surface with nonempty boundary to itself is induced by a homeomorphism of the surface. We also prove that the automorphism group of the arc complex is isomorphic to the quotient of the mapping class group of the surface by its center.
"Injective simplicial maps of the complex of arcs on nonorientable surfaces." Algebr. Geom. Topol. 9 (4) 2055 - 2077, 2009. https://doi.org/10.2140/agt.2009.9.2055