We describe a uniform approach to two known graph drawing results including Gioan’s theorem, stating that any two good drawings of a complete graph with the same rotation system are isomorphic up to Reidemeister moves of type 3, and a characterization of pseudolinear drawings of the complete graph via an excluded configuration: a bad \(K_4\). Our approach yields a new and short self-contained proof of Gioan’s theorem, and a short proof of the pseudolinearity characterization using a previous result. As a bonus we obtain an extension of Gioan’s theorem to the family of graphs \(K_n-M\), where M is a non-perfect matching in \(K_n\), \(n \ge 5\).

我们描述了一种统一的方法来处理包括Gioan定理在内的两个已知图形的绘图结果，指出具有相同旋转系统的完整图形的任何两个良好图形都是同构的，直到类型3的Reidemeister运动为止，并且表征了完整图形的伪线性图形通过排除的配置：错误的\（K_4 \）。我们的方法产生了Gioan定理的新的短的自包含证明，以及使用先前的结果得到的伪线性特征的短证明。作为奖励，我们获得了Gioan定理的扩展到图族\（K_n-M \），其中M是\（K_n \），\（n \ ge 5 \）中的非完美匹配 。