The Complexity of Color-constrained Paths in Semicomplete Multipartite Digraphs

Every semicomplete multipartite digraph contains a quasi-Hamiltonian path, but the problem of finding a quasi-Hamiltonian path with prescribed start and end vertex is NP-complete even when restricted to semicomplete multipartite digraphs with independence number exactly 3. Bang-Jensen, Wang and Yeo (arXiv 2024) showed that deciding the presence of a quasi-Hamiltonian cycle which does not contain at least one vertex from each color class is NP-complete. Similarly, deciding the presence of a quasi-Hamiltonian cycle which intersects every part exactly once is also NP-complete as shown in the same work.
In this paper, we continue the study of paths with constraints on the number of covered vertices from each color class. We consider the problem of finding a path with prescribed start and end vertex that contains at least $a$ and at most $b$ vertices from each color class where all color classes have size exactly $\alpha$. This unifies the Hamiltonian path problem, the quasi-Hamiltonian path problem and the path-version of the cycle problems mentioned above, among other problems. Using Schaefer's dichotomy theorem, we classify the complexity of almost all problems in our framework. Notable open problems are the Hamiltonian path problem on semicomplete multipartite digraphs as well as the quasi-Hamiltonian path problem restricted to semicomplete multipartite digraphs with independence number 2.
We then investigate the quasi-Hamiltonian path problem restricted to semicomplete multipartite digraphs with independence number 2. We generalize sufficient criteria for Hamiltonian $(s,t)$-paths in semicomplete digraphs to sufficient criteria for quasi-Hamiltonian $(s,t)$-paths in this class. Although this does not settle the problem, the initial results suggest that this special case may be solvable in polynomial time.