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.

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