Drawings of Complete Multipartite Graphs Up to Triangle Flips

TL;DR

This work extends Gioan's triangle-flip transformability to complete multipartite graphs by showing that two simple drawings with identical extended rotation systems (ERS) can be transformed into one another via triangle flips. It provides two proofs—one concise and another algorithmic—that yield a polynomial-time transformation with an upper bound of $O(n^{16})$ flips and demonstrates a matching $\Omega(n^{6})$ lower bound in worst cases. A Carathéodory-type theorem for these graphs underpins the second proof, and the results include tightness phenomena illustrating the limits of ERS-based equivalence when slightly modifying the edge set. The paper also situates these findings in the broader context of rotation systems, flip graphs, and crossing structures, and outlines open questions for complete characterization and bound refinement.

Abstract

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 Carathéodory-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.

Figures (13)

• Figure 1: A sketch of a triangle flip.
• Figure 2: Two simple drawings of $K_{3,3}$ with the same crossing edge pairs and same rotations at all vertices but different rotations at all crossings involving the edge $b_1r_3$ and hence different ERSs.
• Figure 3: Two drawings of $K_{3,3}$ that have same ERS but are not strongly isomorphic (because $ux$ crosses $vy$ and $wz$ in different order). The shaded tricell is an invertible triangle with different parities in the two drawings.
• Figure 4: Two drawings of a path with the same ERS, but the order of crossings along the edge $cd$ differs, thus, the drawings are not strongly isomorphic. Neither drawing contains any tricell to flip.
• Figure 6: Two drawings of $K_{m,n}$ minus two adjacent edges $b_1r_2$ and $b_1r_3$ (drawn as dashed lines) that have the same ERS but cannot be transformed into each other via triangle flips.

Theorems & Definitions (29)

• Theorem 1
• Theorem 2
• Theorem 3: Carathéodory-type theorem for simple drawings of complete multipartite graphs
• Lemma 3
• Lemma 3
• Theorem 5
• Theorem 6
• Lemma 6
• Lemma 6
• Theorem 6: Carathéodory-type theorem for simple drawings of complete multipartite graphs