The Complexity of Contracting Bipartite Graphs into Small Cycles

TL;DR

This paper investigates the computational complexity of transforming bipartite graphs into small cycles via edge contractions, proving NP-completeness for cycles of length 4 and 5, and thus resolving open problems in the field.

Contribution

It establishes NP-completeness of C4- and C5-Contractibility on bipartite graphs, advancing understanding of graph contraction problems.

Findings

C4-Contractibility is NP-complete on bipartite graphs.

C5-Contractibility is NP-complete on bipartite graphs.

Resolved open problems regarding small cycle contraction complexity.

Abstract

For a positive integer $\ell \geq 3$, the $C_\ell$-Contractibility problem takes as input an undirected simple graph $G$ and determines whether $G$ can be transformed into a graph isomorphic to $C_\ell$ (the induced cycle on $\ell$ vertices) using only edge contractions. Brouwer and Veldman [JGT 1987] showed that $C_4$-Contractibility is NP-complete in general graphs. It is easy to verify that $C_3$-Contractibility is polynomial-time solvable. Dabrowski and Paulusma [IPL 2017] showed that $C_{\ell}$-Contractibility is \NP-complete\ on bipartite graphs for $\ell = 6$ and posed as open problems the status of the problem when $\ell$ is 4 or 5. In this paper, we show that both $C_5$-Contractibility and $C_4$-Contractibility are NP-complete on bipartite graphs.