# Contracting Bipartite Graphs to Paths and Cycles

## TL;DR

This paper investigates the computational complexity of contracting bipartite graphs into paths and cycles, proving NP-completeness for certain cases and strengthening existing results.

## Contribution

It improves previous NP-completeness results for contracting bipartite graphs into paths, showing NP-completeness for k=5, and proves NP-completeness for contracting to C6, impacting cyclicity determination.

## Key findings

- Contracting bipartite graphs to P5 is NP-complete.
- Contracting bipartite graphs to C6 is NP-complete.
- Determining bipartite graph cyclicity is NP-hard.

## Abstract

Testing if a given graph $G$ contains the $k$-vertex path $P_k$ as a minor or as an induced minor is trivial for every fixed integer $k\geq 1$. However, the situation changes for the problem of checking if a graph can be modified into $P_k$ 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, L\'{e}v\^{e}que 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 $C_6$ 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.