Separable Drawings: Extendability and Crossing-Free Hamiltonian Cycles
Oswin Aichholzer, Joachim Orthaber, Birgit Vogtenhuber
TL;DR
The paper introduces separable drawings, a broad relaxation of pseudospherical drawings where each edge has a witness curve that meets every other edge at most once, and uses this framework to study two classic questions: extendability to simple drawings of $K_n$ and the existence of crossing-free Hamiltonian substructures. It proves that any separable drawing of a graph on $n$ vertices extends to a simple drawing of $K_n$, and that separable drawings of $K_n$ contain a crossing-free Hamiltonian cycle and are plane Hamiltonian connected. It also shows that generalized convex drawings and 2-page book drawings are separable, placing separable drawings strictly above these classes and strengthening recent results on plane Hamiltonicity. Additionally, the work develops a rotation-system perspective, with flips and triangle mutations linking separability to realizability, and analyzes recognition complexity (polynomial-time for simple drawings of $K_n$, NP-complete for general graphs).
Abstract
Generalizing pseudospherical drawings, we introduce a new class of simple drawings, which we call separable drawings. In a separable drawing, every edge can be closed to a simple curve that intersects each other edge at most once. Curves of different edges might interact arbitrarily. Most notably, we show that (1) every separable drawing of any graph on $n$ vertices in the plane can be extended to a simple drawing of the complete graph $K_{n}$, (2) every separable drawing of $K_{n}$ contains a crossing-free Hamiltonian cycle and is plane Hamiltonian connected, and (3) every generalized convex drawing and every 2-page book drawing is separable. Further, the class of separable drawings is a proper superclass of the union of generalized convex and 2-page book drawings. Hence, our results on plane Hamiltonicity extend recent work on generalized convex drawings by Bergold et al. (SoCG 2024).