For a drawing of a labeled graph, the rotation of a vertex or crossing is the cyclic order of its incident edges, represented by the labels of their other endpoints. The extended rotation system (ERS) of the drawing is the collection of the rotations of all vertices and crossings. A drawing is simple if each pair of edges has at most one common point. Gioan's Theorem states that for any two simple drawings of the complete graph $K_n$ with the same crossing edge pairs, one drawing can be transformed into the other by a sequence of triangle flips (a.k.a. Reidemeister moves of Type 3). This operation refers to the act of moving one edge of a triangular cell formed by three pairwise crossing edges over the opposite crossing of the cell, via a local transformation. We investigate to what extent Gioan-type theorems can be obtained for wider classes of graphs. A necessary (but in general not sufficient) condition for two drawings of a graph to be transformable into each other by a sequence of triangle flips is that they have the same ERS. As our main result, we show that for the large class of complete multipartite graphs, this necessary condition is in fact also sufficient. We present two different proofs of this result, one of which is shorter, while the other one yields a polynomial time algorithm for which the number of needed triangle flips for graphs on $n$ vertices is bounded by $O(n^{16})$. The latter proof uses a Carath\'eodory-type theorem for simple drawings of complete multipartite graphs, which we believe to be of independent interest. Moreover, we show that our Gioan-type theorem for complete multipartite graphs is essentially tight in the sense that having the same ERS does not remain sufficient when removing or adding very few edges.

翻译：对于带标签图的画图，顶点或交叉点的旋转是指其关联边的循环顺序，由这些边另一端点标签表示。画图的扩展旋转系统（ERS）是顶点和交叉点旋转的集合。若每对边至多有一个公共点，则画图是简单的。Gioan定理指出：对于完全图 $K_n$ 的任意两个具有相同交叉边对的简单画图，其中一个画图可通过一系列三角翻转（即第三类Reidemeister移动）转化为另一个。此操作指通过局部变换，将三条两两交叉边形成的三角形单元中的一条边移至该单元对侧交叉点上方。我们探究对于更广泛的图类，Gioan型定理能在何种程度上成立。要使两个图画图可通过三角翻转序列相互转化，一个必要条件（通常不充分）是它们具有相同的ERS。作为主要结果，我们证明对于完全多部图这一大类图，这个必要条件实际上也是充分的。我们给出该结果的两个不同证明：一个较短，另一个则提供了多项式时间算法，其对 $n$ 个顶点的图所需三角翻转次数以 $O(n^{16})$ 为界。后一个证明使用了完全多部图简单画图的Carathéodory型定理，我们认为该定理本身具有独立意义。此外，我们证明完全多部图的Gioan型定理本质上是紧的，即当移除或添加极少量边时，具有相同ERS不再是充分条件。