Separable Drawings: Extendability and Crossing-Free Hamiltonian Cycles

TL;DR

This paper introduces separable drawings, a new class of simple graph drawings, and proves they can be extended to complete graphs with crossing-free Hamiltonian cycles, unifying several known drawing classes.

Contribution

The paper defines separable drawings and proves they can be extended to complete graphs with crossing-free Hamiltonian cycles, extending previous results to a broader class.

Findings

Separable drawings can be extended to simple drawings of complete graphs.

Every separable drawing of K_n contains a crossing-free Hamiltonian cycle.

Generalized convex and 2-page book drawings are subclasses of separable drawings.

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).