🤖 AI Summary
This paper investigates the computational complexity of determining Hamiltonian paths, Hamiltonian cycles, and generalized Hamiltonian-ℓ-connectivity—i.e., the existence of ℓ vertex-disjoint paths covering all vertices, given ℓ prescribed endpoint pairs—in graph classes with bounded independence number. We develop a unified framework combining structural graph theory and dynamic programming, integrating techniques such as bounded independent set enumeration, block decomposition, and pairing-based path construction. We establish, for the first time, that all three problems are polynomial-time solvable on graphs G satisfying α(G) ≤ k, for any fixed k, ℓ ∈ ℕ, with runtime O(n^{f(k,ℓ)}). This resolves long-standing open questions—including the polynomial-time solvability on 4K₁-free graphs—and substantially extends the known boundary of polynomial-time tractability for Hamiltonicity-related problems.
📝 Abstract
A Hamiltonian path (a Hamiltonian cycle) in a graph is a path (a cycle, respectively) that traverses all of its vertices. The problems of deciding their existence in an input graph are well-known to be NP-complete, in fact, they belong to the first problems shown to be computationally hard when the theory of NP-completeness was being developed. A lot of research has been devoted to the complexity of Hamiltonian path and Hamiltonian cycle problems for special graph classes, yet only a handful of positive results are known. The complexities of both of these problems have been open even for $4K_1$-free graphs, i.e., graphs of independence number at most $3$. We answer this question in the general setting of graphs of bounded independence number. We also consider a newly introduced problem called emph{Hamiltonian-$ell$-Linkage} which is related to the notions of a path cover and of a linkage in a graph. This problem asks if given $ell$ pairs of vertices in an input graph can be connected by disjoint paths that altogether traverse all vertices of the graph. For $ell=1$, Hamiltonian-1-Linkage asks for existence of a Hamiltonian path connecting a given pair of vertices. Our main result reads that for every pair of integers $k$ and $ell$, the Hamiltonian-$ell$-Linkage problem is polynomial time solvable for graphs of independence number not exceeding $k$.