  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.
