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**Published in:** LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

We investigate the problem of modifying a graph into a connected graph in which the degree of each vertex satisfies a prescribed parity constraint. Let ea, ed and vd denote the operations edge addition, edge deletion and vertex deletion respectively. For any S subseteq {ea,ed,vd}, we define Connected Degree Parity Editing (S) (CDPE(S)) to be the problem that takes as input a graph G, an integer k and a function delta: V(G) -> {0,1}, and asks whether G can be modified into a connected graph H with d_H(v) = delta(v)(mod 2) for each v in V(H), using at most k operations from S. We prove that (*) if S={ea} or S={ea,ed}, then CDPE(S) can be solved in polynomial time; (*) if {vd} subseteq S subseteq {ea,ed,vd}, then CDPE(S) is NP-complete and W-hard when parameterized by k, even if delta = 0.
Together with known results by Cai and Yang and by Cygan, Marx, Pilipczuk, Pilipczuk and Schlotter, our results completely classify the classical and parameterized complexity of the CDPE(S) problem for all S subseteq {ea,ed,vd}. We obtain the same classification for a natural variant of the cdpe(S) problem on directed graphs, where the target is a weakly connected digraph in which the difference between the in- and out-degree of every vertex equals a prescribed value.
As an important implication of our results, we obtain polynomial-time algorithms for Eulerian Editing problem and its directed variant. To the best of our knowledge, the only other natural non-trivial graph class H for which the H-Editing problem is known to be polynomial-time solvable is the class of split graphs.

Konrad K. Dabrowski, Petr A. Golovach, Pim van 't Hof, and Daniel Paulusma. Editing to Eulerian Graphs. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 97-108, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{dabrowski_et_al:LIPIcs.FSTTCS.2014.97, author = {Dabrowski, Konrad K. and Golovach, Petr A. and van 't Hof, Pim and Paulusma, Daniel}, title = {{Editing to Eulerian Graphs}}, booktitle = {34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)}, pages = {97--108}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-77-4}, ISSN = {1868-8969}, year = {2014}, volume = {29}, editor = {Raman, Venkatesh and Suresh, S. P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.97}, URN = {urn:nbn:de:0030-drops-48356}, doi = {10.4230/LIPIcs.FSTTCS.2014.97}, annote = {Keywords: Eulerian graphs, graph editing, polynomial algorithm} }

Document

**Published in:** LIPIcs, Volume 13, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)

We initiate the study of the Bipartite Contraction problem from the perspective of parameterized complexity. In this problem we are given a graph G on n vertices and an integer k, and the task is to determine whether we can obtain a bipartite graph from G by a sequence of at most k edge contractions. Our main result is an f(k) n^{O(1)} time algorithm for Bipartite Contraction. Despite a strong resemblance between Bipartite Contraction and the classical Odd Cycle Transversal (OCT) problem, the methods developed to tackle OCT do not seem to be directly applicable to Bipartite Contraction. To obtain our result, we combine several techniques and concepts that are central in parameterized complexity: iterative compression, irrelevant vertex, and important separators. To the best of our knowledge, this is the first time the irrelevant vertex technique and the concept of important separators are applied in unison. Furthermore, our algorithm may serve as a comprehensible example of the usage of the irrelevant vertex technique.

Pinar Heggernes, Pim van 't Hof, Daniel Lokshtanov, and Christophe Paul. Obtaining a Bipartite Graph by Contracting Few Edges. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011). Leibniz International Proceedings in Informatics (LIPIcs), Volume 13, pp. 217-228, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2011)

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@InProceedings{heggernes_et_al:LIPIcs.FSTTCS.2011.217, author = {Heggernes, Pinar and van 't Hof, Pim and Lokshtanov, Daniel and Paul, Christophe}, title = {{Obtaining a Bipartite Graph by Contracting Few Edges}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2011)}, pages = {217--228}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-34-7}, ISSN = {1868-8969}, year = {2011}, volume = {13}, editor = {Chakraborty, Supratik and Kumar, Amit}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2011.217}, URN = {urn:nbn:de:0030-drops-33579}, doi = {10.4230/LIPIcs.FSTTCS.2011.217}, annote = {Keywords: fixed parameter tractability, graph modification problems, edge contractions, bipartite graphs} }

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