Dynamic Generators of Topologically Embedded Graphs

TL;DR

This paper introduces a dynamic data structure for efficiently maintaining graph embeddings on surfaces of arbitrary genus, supporting various topological and combinatorial operations with improved amortized update times.

Contribution

It presents a novel data structure that supports dynamic updates of graph embeddings on surfaces, including genus changes, and improves related separator and tree-decomposition algorithms.

Findings

Supports edge insertion, deletion, and dual operations efficiently.

Enables testing of surface orientability in logarithmic time.

Improves constant factors in separator theorems for low-genus graphs.

Abstract

We provide a data structure for maintaining an embedding of a graph on a surface (represented combinatorially by a permutation of edges around each vertex) and computing generators of the fundamental group of the surface, in amortized time O(log n + log g(log log g)^3) per update on a surface of genus g; we can also test orientability of the surface in the same time, and maintain the minimum and maximum spanning tree of the graph in time O(log n + log^4 g) per update. Our data structure allows edge insertion and deletion as well as the dual operations; these operations may implicitly change the genus of the embedding surface. We apply similar ideas to improve the constant factor in a separator theorem for low-genus graphs, and to find in linear time a tree-decomposition of low-genus low-diameter graphs.