The Path Contraction and Cycle Contraction problems take as input an undirected graph $G$ with $n$ vertices, $m$ edges and an integer $k$ and determine whether one can obtain a path or a cycle, respectively, by performing at most $k$ edge contractions in $G$. We revisit these NP-complete problems and prove the following results. Path Contraction admits an algorithm running in $\mathcal{O}^*(2^{k})$ time. This improves over the current algorithm known for the problem [Algorithmica 2014]. Cycle Contraction admits an algorithm running in $\mathcal{O}^*((2 + \epsilon_{\ell})^k)$ time where $0 < \epsilon_{\ell} \leq 0.5509$ is inversely proportional to $\ell = n - k$. Central to these results is an algorithm for a general variant of Path Contraction, namely, Path Contraction With Constrained Ends. We also give an $\mathcal{O}^*(2.5191^n)$-time algorithm to solve the optimization version of Cycle Contraction. Next, we turn our attention to restricted graph classes and show the following results. Path Contraction on planar graphs admits a polynomial-time algorithm. Path Contraction on chordal graphs does not admit an algorithm running in time $\mathcal{O}(n^{2-\epsilon} \cdot 2^{o(tw)})$ for any $\epsilon > 0$, unless the Orthogonal Vectors Conjecture fails. Here, $tw$ is the treewidth of the input graph. The second result complements the $\mathcal{O}(nm)$-time, i.e., $\mathcal{O}(n^2 \cdot tw)$-time, algorithm known for the problem [Discret. Appl. Math. 2014].

翻译：路径收缩（Path Contraction）和圈收缩（Cycle Contraction）问题输入为一个具有$n$个顶点、$m$条边的无向图$G$以及一个整数$k$，要求判定是否可以通过对$G$执行至多$k$次边收缩操作，分别得到一条路径或一个圈。我们重新审视这些NP完全问题，并证明以下结果：路径收缩存在一个运行时间为$\mathcal{O}^*(2^{k})$的算法，这改进了该问题当前已知算法（参见Algorithmica 2014）的性能；圈收缩存在一个运行时间为$\mathcal{O}^*((2 + \epsilon_{\ell})^k)$的算法，其中$0 < \epsilon_{\ell} \leq 0.5509$与$\ell = n - k$成反比。这些结果的核心是路径收缩的一个广义变体——带端点约束的路径收缩（Path Contraction With Constrained Ends）的算法。我们还给出一个$\mathcal{O}^*(2.5191^n)$时间的算法，用于求解圈收缩的优化版本。接下来，我们转向受限图类的研究，并展示以下结果：平面图上的路径收缩存在多项式时间算法；弦图上的路径收缩不存在运行时间为$\mathcal{O}(n^{2-\epsilon} \cdot 2^{o(tw)})$的算法（其中$\epsilon > 0$），除非正交向量猜想（Orthogonal Vectors Conjecture）不成立。此处$tw$表示输入图的树宽。第二个结果补充了该问题已知的$\mathcal{O}(nm)$时间（即$\mathcal{O}(n^2 \cdot tw)$时间）算法（参见Discret. Appl. Math. 2014）。