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.

翻译：对于正整数 $\ell \geq 3$，$C_\ell$-可收缩性问题以一个无向简单图 $G$ 作为输入，并判定 $G$ 是否能够仅通过边收缩操作变换为一个与 $C_\ell$（包含 $\ell$ 个顶点的导出环）同构的图。Brouwer 和 Veldman [JGT 1987] 证明了在一般图中 $C_4$-可收缩性是 NP 完全的。容易验证 $C_3$-可收缩性存在多项式时间算法。Dabrowski 和 Paulusma [IPL 2017] 证明了在二分图上，当 $\ell = 6$ 时 $C_{\ell}$-可收缩性是 NP 完全的，并提出了当 $\ell$ 为 4 或 5 时该问题的复杂性作为开放问题。本文中，我们证明了在二分图上，$C_5$-可收缩性与 $C_4$-可收缩性均为 NP 完全问题。