Hamiltonian path and Hamiltonian cycle are solvable in polynomial time in graphs of bounded independence number

TL;DR

The paper addresses the longstanding NP-hardness of Hamiltonian path and cycle problems by identifying a broad graph class—those with bounded independence number—where these problems, and their natural generalization Hamiltonian-$\ell$-Linkage, become solvable in polynomial time. The authors develop a recursive, cut-based algorithm underpinned by a new Hamiltonian-$\ell$-linkage framework and a scenario machinery that coherently connects disjoint components to cover all vertices. A key theoretical contribution is showing that for any fixed $k$ and $\ell$, the Hamiltonian-$\ell$-Linkage problem is polynomial-time solvable on graphs with $\alpha(G)<k$, with explicit reductions to highly connected cases and carefully controlled branching on small vertex cuts. They further relate these linkage results to $L(2,1)$-labelling, derive fixed-parameter tractability implications, and discuss the limitations of extending the results to graphs with bounded tree independence number. The work advances our understanding of the boundary between easy and hard Hamiltonian-type problems and yields practical polynomial-time algorithms for a broad, natural graph class.

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 $k$.

Figures (1)

• Figure 2: An example of a scenario in the proof of Theorem \ref{['thm:Ham-linkage-poly']}.

Theorems & Definitions (23)

• Theorem 1
• Proposition 2
• Proposition 3
• Definition 4
• Proposition 5
• Definition 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10