Contracting bipartite graphs to paths and cycles

- Determining if a bipartite graph can be contracted to the 5-vertex path is NP-complete.
- Determining if a bipartite graph can be contracted to the 6-vertex cycle is NP-complete.
- Computing the cyclicity of a bipartite graph is NP-hard.

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.