On Dynamic Parameterized $ k $-Path

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 2O(d2), 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 2O(k2). In the second case, the amortized update time is 2O(k4)+O(klogn). In both cases we assume access to a dictionary on the edges of G.