We study the \textsc{Labeled Contractibility} problem, where the input consists of two vertex-labeled graphs $G$ and $H$, and the goal is to determine whether $H$ can be obtained from $G$ via a sequence of edge contractions. Lafond and Marchand~[WADS 2025] initiated the parameterized complexity study of this problem, showing it to be \(\W[1]\)-hard when parameterized by the number \(k\) of allowed contractions. They also proved that the problem is fixed-parameter tractable when parameterized by the tree-width \(\tw\) of \(G\), via an application of Courcelle's theorem resulting in a non-constructive algorithm. In this work, we present a constructive fixed-parameter algorithm for \textsc{Labeled Contractibility} with running time \(2^{\mathcal{O}(\tw^2)} \cdot |V(G)|^{\mathcal{O}(1)}\). We also prove that unless the Exponential Time Hypothesis (Ð) fails, it does not admit an algorithm running in time \(2^{o(\tw^2)} \cdot |V(G)|^{\mathcal{O}(1)}\). This result adds \textsc{Labeled Contractibility} to a small list of problems that admit such a lower bound and matching algorithm. We further strengthen existing hardness results by showing that the problem remains \NP-complete even when both input graphs have bounded maximum degree. We also investigate parameterizations by \((k + δ(G))\) where \(δ(G)\) denotes the degeneracy of \(G\), and rule out the existence of subexponential-time algorithms. This answers question raised in Lafond and Marchand~[WADS 2025]. We additionally provide an improved \FPT\ algorithm with better dependence on \((k + δ(G))\) than previously known. Finally, we analyze a brute-force algorithm for \textsc{Labeled Contractibility} with running time \(|V(H)|^{\mathcal{O}(|V(G)|)}\), and show that this running time is optimal under Ð.

翻译：我们研究\textsc{带标签可收缩性}问题，其输入包含两个顶点带标签的图$G$和$H$，目标是判断$H$能否通过一系列边收缩操作从$G$得到。Lafond和Marchand~[WADS 2025]开创了该问题的参数化复杂性研究，证明当以允许的收缩次数$k$为参数时，该问题是\(\W[1]\)-难的。他们还通过应用Courcelle定理证明了当以$G$的树宽\(\tw\)为参数时，该问题是固定参数可解的，但得到的是非构造性算法。本文提出了一种构造性固定参数算法求解\textsc{带标签可收缩性}问题，其运行时间为\(2^{\mathcal{O}(\tw^2)} \cdot |V(G)|^{\mathcal{O}(1)}\)。我们同时证明，除非指数时间假设（Ð）不成立，否则该问题不存在\(2^{o(\tw^2)} \cdot |V(G)|^{\mathcal{O}(1)}\)时间的算法。该结果使\textsc{带标签可收缩性}成为少数同时具备此类下界与匹配算法的问题之一。我们进一步强化了现有硬度结果，证明即使两个输入图都具有有界最大度时，该问题仍然是\NP-完全的。我们还研究了以\((k + δ(G))\)为参数的情况（其中\(δ(G)\)表示$G$的退化度），并排除了亚指数时间算法的存在性，这回答了Lafond和Marchand~[WADS 2025]中提出的问题。此外，我们提出了一种改进的\FPT\算法，其在\((k + δ(G))\)上的依赖度优于已知结果。最后，我们分析了\textsc{带标签可收缩性}的一个暴力算法，其运行时间为\(|V(H)|^{\mathcal{O}(|V(G)|)}\)，并证明在Ð假设下该运行时间是最优的。