Conference paper
On Dynamic α+ 1 Arboricity Decomposition and Out-Orientation
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 O(n3/4) 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 O (n5/7), 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 O(n1/2) time.
Language: | English |
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Publisher: | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |
Year: | 2022 |
Pages: | 1-15 |
Proceedings: | 47<sup>th</sup> International Symposium on Mathematical Foundations of Computer Science |
Series: | Leibniz International Proceedings in Informatics, Lipics |
ISBN: | 3959772564 and 9783959772563 |
ISSN: | 18688969 |
Types: | Conference paper |
DOI: | 10.4230/LIPIcs.MFCS.2022.34 |
ORCIDs: | 0000-0001-6997-9251 , Christiansen, Aleksander B.G. , Rotenberg, Eva and Thomassen, Carsten |