Efficient Enumeration of Drawings and Combinatorial Structures for Maximal Planar Graphs

TL;DR

This paper introduces efficient algorithms for enumerating key combinatorial structures and graph drawings in maximal planar graphs, establishing bijections and achieving linear time and space complexity.

Contribution

It presents novel enumeration algorithms for canonical orientations, and establishes bijections between orientations, orderings, and drawings with linear efficiency.

Findings

Enumeration algorithms run in O(n) time and space

Bijections between orientations, orderings, and drawings are established

Algorithms efficiently enumerate structures for maximal planar graphs

Abstract

We propose efficient algorithms for enumerating the notorious combinatorial structures of maximal planar graphs, called canonical orderings and Schnyder woods, and the related classical graph drawings by de Fraysseix, Pach, and Pollack [Combinatorica, 1990] and by Schnyder [SODA, 1990], called canonical drawings and Schnyder drawings, respectively. To this aim (i) we devise an algorithm for enumerating special $e$-bipolar orientations of maximal planar graphs, called canonical orientations; (ii) we establish bijections between canonical orientations and canonical drawings, and between canonical orientations and Schnyder drawings; and (iii) we exploit the known correspondence between canonical orientations and canonical orderings, and the known bijection between canonical orientations and Schnyder woods. All our enumeration algorithms have $O(n)$ setup time, space usage, and delay between any two consecutively listed outputs, for an $n$-vertex maximal planar graph.