{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article6710","name":"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.\r\n\r\nTutte 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.","keywords":["Construction sequence","3-connected graph","nested subdivisions","inductive characterization","3-connectedness","certifying algorithm"],"author":{"@type":"Person","name":"Schmidt, Jens M.","givenName":"Jens M.","familyName":"Schmidt"},"position":55,"pageStart":633,"pageEnd":644,"dateCreated":"2010-03-09","datePublished":"2010-03-09","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by-nd\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Schmidt, Jens M.","givenName":"Jens M.","familyName":"Schmidt"},"copyrightYear":"2010","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.STACS.2010.2491","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6208","volumeNumber":5,"name":"27th International Symposium on Theoretical Aspects of Computer Science","dateCreated":"2010-03-09","datePublished":"2010-03-09","editor":[{"@type":"Person","name":"Marion, Jean-Yves","givenName":"Jean-Yves","familyName":"Marion"},{"@type":"Person","name":"Schwentick, Thomas","givenName":"Thomas","familyName":"Schwentick"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article6710","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6208"}}}