Simultaneous Embedding of a Planar Graph and Its Dual on the Grid

TL;DR

This paper presents a linear-time algorithm for embedding a planar graph and its dual simultaneously on a small grid, ensuring primal-dual edges cross only each other and vertices are placed inside their faces.

Contribution

It introduces a novel linear-time method for simultaneous embedding of a planar graph and its dual on a small integer grid with natural face placement and crossing constraints.

Findings

Embedding achieved on a (2n-2) by (2n-2) grid

Algorithm runs in linear time

Embeds 3-connected planar graphs and their duals with correct crossings

Abstract

Traditional representations of graphs and their duals suggest the requirement that the dual vertices be placed inside their corresponding primal faces, and the edges of the dual graph cross only their corresponding primal edges. We consider the problem of simultaneously embedding a planar graph and its dual into a small integer grid such that the edges are drawn as straight-line segments and the only crossings are between primal-dual pairs of edges. We provide a linear-time algorithm that simultaneously embeds a 3-connected planar graph and its dual on a (2n-2) by (2n-2) integer grid, where n is the total number of vertices in the graph and its dual. Furthermore our embedding algorithm satisfies the two natural requirements mentioned above.