The connection between Hamiltonicity and the independence numbers of graphs has been a fundamental aspect of Graph Theory since the seminal works of the 1960s. This paper presents a novel algorithmic perspective on these classical problems. Our contributions are twofold. First, we establish that a wide array of problems in undirected graphs, encompassing problems such as Hamiltonian Path and Cycle, Path Cover, Largest Linkage, and Topological Minor Containment are fixed-parameter tractable (FPT) parameterized by the independence number of a graph. To the best of our knowledge, these results mark the first instances of FPT problems for such parameterization. Second, we extend the algorithmic scope of the Gallai-Milgram theorem. The original theorem by Gallai and Milgram, asserts that for a graph G with the independence number \alpha(G), the vertex set of G can be covered by at most \alpha(G) vertex-disjoint paths. We show that determining whether a graph can be covered by fewer than \alpha(G) - k vertex-disjoint paths is FPT parameterized by k. Notably, the independence number parameterization, which describes graph's density, departs from the typical flow of research in parameterized complexity, which focuses on parameters describing graph's sparsity, like treewidth or vertex cover.

翻译：自20世纪60年代开创性研究以来，哈密顿性与图的独立数之间的联系一直是图论的基础问题。本文针对这些经典问题提出了全新的算法视角。我们的贡献体现在两方面。首先，我们证明无向图中的一系列问题（包括哈密顿路径与圈问题、路径覆盖、最大链路问题以及拓扑子式包含问题）在图的独立数参数化下均是固定参数可解的（FPT）。据我们所知，这些结果首次证明了此类参数化问题的FPT性质。其次，我们扩展了Gallai-Milgram定理的算法适用范围。Gallai和Milgram的原始定理指出：对于独立数为α(G)的图G，其顶点集至多可由α(G)条顶点不相交路径覆盖。我们证明：判断图是否能被少于α(G)-k条顶点不相交路径覆盖这一问题，在k参数化下属于FPT。值得注意的是，描述图稠密度的独立数参数化，突破了参数化复杂度研究中以树宽或顶点覆盖等稀疏性参数为主流的研究范式。