Polygon-Universal Graphs
Tim Ophelders, Ignaz Rutter, Bettina Speckmann, Kevin Verbeek

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 of a graph and a cycle in together with a simple polygon , is there a straight-line drawing of inside which maps to ? We say that such a drawing of respects . We fully characterize those instances which are polygon-universal, that is, they have a drawing that respects for any simple (not necessarily convex) polygon . 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 is planar, that is, if there exists a planar drawing of with on the outer face, we show that the same conditions guarantee for every simple polygon the existence of a…
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