Let [Formula: see text] be a compact, connected, orientable surface of genus [Formula: see text] with [Formula: see text] boundary components with [Formula: see text], [Formula: see text]. Let [Formula: see text] be the nonseparating curve graph, [Formula: see text] be the curve graph and [Formula: see text] be the Hatcher–Thurston graph of [Formula: see text]. We prove that if [Formula: see text] is an edge-preserving map, then [Formula: see text] is induced by a homeomorphism of [Formula: see text]. We prove that if [Formula: see text] is an edge-preserving map, then [Formula: see text] is induced by a homeomorphism of [Formula: see text]. We prove that if [Formula: see text] is closed and [Formula: see text] is a rectangle preserving map, then [Formula: see text] is induced by a homeomorphism of [Formula: see text]. We also prove that these homeomorphisms are unique up to isotopy when [Formula: see text].

中文翻译：

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非分离曲线图的边缘保留图、Hatcher-Thurston 图的曲线图和矩形保留图

令 [Formula: see text] 是 [Formula: see text] 与 [Formula: see text] 的边界组件与 [Formula: see text], [Formula: see text] 的一个紧凑的、连接的、可定向的表面。令[公式：见文]为非分离曲线图，[公式：见文]为曲线图，[公式：见文]为[公式：见文]的Hatcher-Thurston图。我们证明如果[Formula: see text] 是一个保边映射，那么[Formula: see text] 是由[Formula: see text] 的同胚推导出来的。我们证明如果[Formula: see text] 是一个保边映射，那么[Formula: see text] 是由[Formula: see text] 的同胚推导出来的。我们证明如果[Formula: see text] 是封闭的并且[Formula: see text] 是一个矩形保留映射，那么[Formula: see text] 是由[Formula: see text] 的同胚推导出来的。