无长导出路径图中k-着色问题的参数化复杂性
Parameterized complexity of $k$-Coloring in graphs with no long induced paths
- Warsaw University of Technology（华沙理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究证明了无长导出路径图中k-着色和3-着色问题的参数化复杂性，解决了两个图论领域的长期开放问题，并基于指数时间假设给出了算法下界。
AI中文摘要：
我们研究当H为线性森林（即路径的不交并）作为导出子图时，H-自由图中k-着色问题的参数化复杂性。我们证明两个困难性结果：1. 当参数为k时，2P₂-自由图中的k-着色问题是W[1]-困难的；2. 当参数为t时，P_t-自由图中的3-着色问题是W[1]-困难的。此外，假设指数时间假设（ETH）成立，这些问题不存在任何可计算函数f，使得能在f(k)·n^{o(k)}和f(t)·n^{o(t/log t)}时间内求解n个顶点的实例。第一个结果以强形式解决了Hoàng、Kamiński、Lozin、Sawada和Shu于《Algorithmica》2010年提出的长期开放问题，第二个结果回答了Golovach、Johnson、Paulusma和Song于《Journal of Graph Theory》2017年提出的问题。
英文摘要：
We study the parameterized complexity of (List) $k$-Coloring in $H$-free graphs, where $H$ is a linear forest, that is, a disjoint union of paths. First, considering $k$ as the parameter, we establish the following: * For any $s \geq 0$, List $k$-Coloring in $(P_4+sP_1)$-free graphs is fixed-parameter tractable (FPT). * $k$-Coloring is W[1]-hard in $2P_2$-free graphs. The second result settles, in a strong form, a long-standing open problem posed by Hoàng, Kamiński, Lozin, Sawada, and Shu [Algorithmica, 2010]. Next, we prove that $k$-Coloring is NP-hard in $(P_4+P_2)$-free graphs. These three findings, together with known results from classical, non-parameterized complexity, yield a complete complexity classification of $k$-Coloring and List $k$-Coloring in $H$-free graphs, parameterized by $k$, into the cases: FPT, XP but W[1]-hard, and paraNP-hard. We also prove that $3$-Coloring is W[1]-hard in $P_t$-free graphs when parameterized by $t$. This answers a question of Golovach, Johnson, Paulusma, and Song [Journal of Graph Theory, 2017]. Finally, as a byproduct of our algorithm for List $k$-Coloring in $(P_4+sP_1)$-free graphs, we show that, for every fixed $s$ and $k$, there are only finitely many $(P_4+sP_1)$-free minimal obstructions to $k$-colorability. This settles a conjecture of Cameron, Hoàng, and Sawada [Disc. Appl. Math., 2022] and completes the dichotomy concerning the finiteness of the family of vertex-$k$-critical $H$-free graphs for every graph $H$ and every $k$.
