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DOI: 10.4230/LIPIcs.STACS.2010.2491
URN: urn:nbn:de:0030-drops-24918
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2010/2491/
Schmidt, Jens M.
Construction Sequences and Certifying 3-Connectedness
Abstract
Given two $3$-connected graphs $G$ and $H$, a \emph{construction sequence} constructs $G$ from $H$ (e.\,g. from the $K_4$) with three basic operations, called the \emph{Barnette-Gr\"unbaum operations}. These operations are known to be able to construct all $3$-connected graphs. We extend this result by identifying every intermediate graph in the construction sequence with a subdivision in $G$ and showing under some minor assumptions that there is still a construction sequence to $G$ when we start from an \emph{arbitrary prescribed} $H$-subdivision. This leads to the first algorithm that computes a construction sequence in time $O(|V(G)|^2)$. As an application, we develop a certificate for the $3$-connectedness of graphs that can be easily computed and verified. Based on this, a certifying test on $3$-connectedness is designed.%Finding certifying algorithms is a major goal for problems where the efficient solutions known are complicated.
Tutte proved that every $3$-connected graph on more than $4$ nodes has a \emph{contractible edge}. Barnette and Gr\"unbaum proved the existence of a \emph{removable edge} in the same setting. We show that the sequence of contractions and the sequence of removals from $G$ to the $K_4$ can be computed in $O(|V|^2)$ time by extending Barnette and Gr\"unbaum's theorem. As an application, we derive a certificate for the $3$-connectedness of graphs that can be easily computed and verified.
BibTeX - Entry
@InProceedings{schmidt:LIPIcs:2010:2491,
author = {Jens M. Schmidt},
title = {{Construction Sequences and Certifying 3-Connectedness}},
booktitle = {27th International Symposium on Theoretical Aspects of Computer Science},
pages = {633--644},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-16-3},
ISSN = {1868-8969},
year = {2010},
volume = {5},
editor = {Jean-Yves Marion and Thomas Schwentick},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2010/2491},
URN = {urn:nbn:de:0030-drops-24918},
doi = {10.4230/LIPIcs.STACS.2010.2491},
annote = {Keywords: Construction sequence, 3-connected graph, nested subdivisions, inductive characterization, 3-connectedness, certifying algorithm}
}
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Keywords: |
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Construction sequence, 3-connected graph, nested subdivisions, inductive characterization, 3-connectedness, certifying algorithm |
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Collection: |
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27th International Symposium on Theoretical Aspects of Computer Science |
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Issue Date: |
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2010 |
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Date of publication: |
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09.03.2010 |
