Drawing Graphs in the Plane with a Prescribed Outer Face and Polynomial Area

TL;DR

This paper presents a polynomial-area algorithm for drawing planar graphs with a prescribed convex outer face, allowing collinear boundary points and solving an open problem in graph drawing.

Contribution

It introduces a new method that guarantees polynomial area for such drawings, improving upon previous exponential-area algorithms and addressing an open problem.

Findings

Produces polynomial-area drawings for planar graphs with prescribed outer face

Allows collinear boundary vertices without overlapping edges

Solves an open problem related to genus-g graph drawings

Abstract

We study the classic graph drawing problem of drawing a planar graph using straight-line edges with a prescribed convex polygon as the outer face. Unlike previous algorithms for this problem, which may produce drawings with exponential area, our method produces drawings with polynomial area. In addition, we allow for collinear points on the boundary, provided such vertices do not create overlapping edges. Thus, we solve an open problem of Duncan et al., which, when combined with their work, implies that we can produce a planar straight-line drawing of a combinatorially-embedded genus-g graph with the graph's canonical polygonal schema drawn as a convex polygonal external face.