Path Cover, Hamiltonicity, and Independence Number: An FPT Perspective

The classic theorem of Gallai and Milgram (1960) asserts that for every graph G, the vertex set of G can be partitioned into at most α(G) vertex-disjoint paths, where α(G) is the maximum size of an independent set in G. The proof of Gallai--Milgram's theorem is constructive and yields a polynomial-time algorithm that computes a covering of G by at most α(G) vertex-disjoint paths. We prove the following algorithmic extension of Gallai--Milgram's theorem for undirected graphs: determining whether an undirected graph can be covered by fewer than α(G) - k vertex-disjoint paths is fixed-parameter tractable (FPT) when parameterized by k. More precisely, we provide an algorithm that, for an n-vertex graph G and an integer parameter k \ge 1, runs in time 2^{k^{O(k^4)}} \cdot n^{O(1)}, and outputs a path cover P of G. Furthermore, it: - either correctly reports that P is a minimum-size path cover, - or outputs, together with P, an independent set of size |P| + k certifying that P contains at most α(G) - k paths. A key subroutine in our algorithm is an FPT algorithm, parameterized by α(G), for deciding whether G contains a Hamiltonian path. This result is of independent interest -- prior to our work, no polynomial-time algorithm for deciding Hamiltonicity was known even for graphs with independence number at most 3. Moreover, the techniques we develop apply to a wide array of problems in undirected graphs, including Hamiltonian Cycle, Path Cover, Largest Linkage, and Topological Minor Containment. We show that all these problems are FPT when parameterized by the independence number of the graph. 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.