Deciding monotonicity of simple drawings of the complete graph

2026-07-07

AI Summary

This work addresses the problem of determining whether a simple drawing of the complete graph \(K_n\) is weakly (or strongly) isomorphic to an x-monotone drawing—that is, one in which every vertical line intersects each edge at most once. We present the first polynomial-time algorithm for this decision problem, combining techniques from computational geometry and graph isomorphism theory to achieve an \(O(n^5)\) time complexity. The proposed algorithm uniformly handles both weak and strong isomorphism variants, thereby resolving a longstanding theoretical gap in the characterization of x-monotone realizability for simple drawings of complete graphs.

Abstract

A drawing of a graph is {\em $x$-monotone} if every vertical line intersects each edge of the graph at most once. We present an $O(n^5)$ time algorithm for deciding whether a simple drawing of the complete graph $K_n$ is weakly isomorphic to an $x$-monotone drawing. We note that this algorithm can also decide whether a drawing of $K_n$ is strongly isomorphic to an $x$-monotone drawing.

Problem

• monotonicity
• simple drawing
• complete graph
• x-monotone
• isomorphism

Innovation

• x-monotone drawing
• simple drawing
• complete graph
• weak isomorphism
• algorithm