Polygon-Universal Graphs

TL;DR

This paper characterizes when a graph with a cycle can be drawn inside any simple polygon while mapping the cycle to the polygon, providing necessary and sufficient conditions and a linear-time construction algorithm.

Contribution

It introduces a complete characterization of polygon-universal graph-cycle instances based on graph and cycle distances, with efficient checking and construction methods.

Findings

Identifies two simple, checkable conditions for polygon-universality.

Proves these conditions are both necessary and sufficient.

Provides a linear-time algorithm for constructing the drawings.

Abstract

We study a fundamental question from graph drawing: given a pair $(G,C)$ of a graph $G$ and a cycle $C$ in $G$ together with a simple polygon $P$, is there a straight-line drawing of $G$ inside $P$ which maps $C$ to $P$? We say that such a drawing of $(G,C)$ respects $P$. We fully characterize those instances $(G,C)$ which are polygon-universal, that is, they have a drawing that respects $P$ for any simple (not necessarily convex) polygon $P$. Specifically, we identify two necessary conditions for an instance to be polygon-universal. Both conditions are based purely on graph and cycle distances and are easy to check. We show that these two conditions are also sufficient. Furthermore, if an instance $(G,C)$ is planar, that is, if there exists a planar drawing of $G$ with $C$ on the outer face, we show that the same conditions guarantee for every simple polygon $P$ the existence of a planar drawing of $(G,C)$ that respects $P$. If $(G,C)$ is polygon-universal, then our proofs directly imply a linear-time algorithm to construct a drawing that respects a given polygon $P$.