When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2022.34
URN: urn:nbn:de:0030-drops-168320
URL: https://drops.dagstuhl.de/opus/volltexte/2022/16832/
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### On Dynamic α + 1 Arboricity Decomposition and Out-Orientation

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### Abstract

A graph has arboricity α if its edges can be partitioned into α forests. The dynamic arboricity decomposition problem is to update a partitioning of the graph’s edges into forests, as a graph undergoes insertions and deletions of edges. We present an algorithm for maintaining partitioning into α+1 forests, provided the arboricity of the dynamic graph never exceeds α. Our algorithm has an update time of Õ(n^{3/4}) when α is at most polylogarithmic in n.
Similarly, the dynamic bounded out-orientation problem is to orient the edges of the graph such that the out-degree of each vertex is at all times bounded. For this problem, we give an algorithm that orients the edges such that the out-degree is at all times bounded by α+1, with an update time of Õ(n^{5/7}), when α is at most polylogarithmic in n. Here, the choice of α+1 should be viewed in the light of the well-known lower bound by Brodal and Fagerberg which establishes that, for general graphs, maintaining only α out-edges would require linear update time.
However, the lower bound by Brodal and Fagerberg is non-planar. In this paper, we give a lower bound showing that even for planar graphs, linear update time is needed in order to maintain an explicit three-out-orientation. For planar graphs, we show that the dynamic four forest decomposition and four-out-orientations, can be updated in Õ(n^{1/2}) time.

### BibTeX - Entry

@InProceedings{christiansen_et_al:LIPIcs.MFCS.2022.34,
author =	{Christiansen, Aleksander B. G. and Holm, Jacob and Rotenberg, Eva and Thomassen, Carsten},
title =	{{On Dynamic \alpha + 1 Arboricity Decomposition and Out-Orientation}},
booktitle =	{47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)},
pages =	{34:1--34:15},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-95977-256-3},
ISSN =	{1868-8969},
year =	{2022},
volume =	{241},
editor =	{Szeider, Stefan and Ganian, Robert and Silva, Alexandra},
publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
}