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Contracting bipartite graphs to paths and cycles.

Dabrowski, K.K. and Paulusma, D. (2017) 'Contracting bipartite graphs to paths and cycles.', Information processing letters., 127 . pp. 37-42.


Testing if a given graph G contains the k -vertex path Pk as a minor or as an induced minor is trivial for every fixed integer k≥1. However, the situation changes for the problem of checking if a graph can be modified into Pk by using only edge contractions. In this case the problem is known to be NP-complete even if k=4. This led to an intensive investigation for testing contractibility on restricted graph classes. We focus on bipartite graphs. Heggernes, van 't Hof, Lévêque and Paul proved that the problem stays NP-complete for bipartite graphs if k=6. We strengthen their result from k=6 to k=5. We also show that the problem of contracting a bipartite graph to the 6-vertex cycle C6 is NP-complete. The cyclicity of a graph is the length of the longest cycle the graph can be contracted to. As a consequence of our second result, determining the cyclicity of a bipartite graph is NP-hard.

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Publisher statement:© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license ( You may copy and distribute the article, create extracts, abstracts and new works from the article, alter and revise the article, text or data mine the article and otherwise reuse the article commercially (including reuse and/or resale of the article) without permission from Elsevier. You must give appropriate credit to the original work, together with a link to the formal publication through the relevant DOI and a link to the Creative Commons user license above. You must indicate if any changes are made but not in any way that suggests the licensor endorses you or your use of the work.
Date accepted:29 June 2017
Date deposited:30 June 2017
Date of first online publication:05 July 2017
Date first made open access:No date available

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