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 optimumheight elimination forest. The data structure achieves worst-case update time 2 O(d 2 ) , 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 d O(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
Given an edge-weighted graph G with a set Q of k terminals, a mimicking network is a graph with the same set of terminals that exactly preserves the sizes of minimum cuts between any partition of the terminals. A natural question in the area of graph compression is to provide as small mimicking networks as possible for input graph G being either an arbitrary graph or coming from a specific graph class.In this note we show an exponential lower bound for cut mimicking networks in planar graphs: there are edge-weighted planar graphs with k terminals that require 2 k−2 edges in any mimicking network. This nearly matches an upper bound of O(k2 2k ) of Rika [SODA 2013, arXiv:1702.05951] and is in sharp contrast with the O(k 2 ) upper bound under the assumption that all terminals lie on a single face [Goranci, Henzinger, Peng, arXiv:1702.01136]. As a side result we show a hard instance for double-
The shortest augmenting path (SAP) algorithm is one of the most classical approaches to the maximum matching and maximum flow problems, e.g., using it Edmonds and Karp (J. ACM 19(2), 248-264 1972) have shown the first strongly polynomial time algorithm for the maximum flow problem. Quite astonishingly, although it has been studied for many years already, this approach is far from being fully understood. This is exemplified by the online bipartite matching problem. In this problem a bipartite graph G = (W B, E) is being revealed online, i.e., in each round one vertex from B with its incident edges arrives. After arrival of this vertex we augment the current matching by using shortest augmenting path. It was conjectured by Chaudhuri et al. (INFOCOM'09) that the total length of all augmenting paths found by SAP is O(n log n). However, no better bound than O(n 2 ) is known The work of all authors was supported by Polish National Science Center grant 2013/11/D/ST6/03100. Additionally, the work of P. Sankowski was partially supported by the project TOTAL (No 677651) that has received funding from ERC. Syst (2018) 62:337-348 even for trees. In this paper we prove an O(n log 2 n) upper bound for the total length of augmenting paths for trees.
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