Efficient fully dynamic elimination forests with applications to detecting long paths and cycles
Jiehua Chen, Wojciech Krzysztof Czerwinski, Yann Disser, Andreas Emil Feldmann, Danny Hermelin, Wojciech Nadara, Marcin Pilipczuk, Michał Pilipczuk, Manuel Sorge, Bartłomiej Wróblewski, Anna Zych-Pawlewicz · Society for Industrial and Applied Mathematics eBooks · 2021
We present a data structure that in a dynamic graph of treedepth at most d, which is modified over time by edge insertions and deletions, maintains an optimum-height elimination forest. The data structure achieves worst-case update time , which matches the best known parameter dependency in the running time of a static fpt algorithm for computing the treedepth of a graph. This improves a result of Dvořák et al. [ESA 2014], who for the same problem achieved update time f(d) for some non-elementary (i.e. tower-exponential) function f. As a by-product, we improve known upper bounds on the sizes of minimal obstructions for having treedepth d from doubly-exponential in d to dO(d). As applications, we design new fully dynamic parameterized data structures for detecting long paths and cycles in general graphs. More precisely, for a fixed parameter k and a dynamic graph G, modified over time by edge insertions and deletions, our data structures maintain answers to the following queries: Does G contain a simple path on k vertices? Does G contain a simple cycle on at least k vertices? In the first case, the data structure achieves amortized update time . In the second case, the amortized update time is . In both cases we assume access to a dictionary on the edges of G.