Hamiltonian prescribed-endpoint paths in extended semicomplete digraphs

Develop a polynomial-time algorithm that, given an extended semicomplete digraph or locally semicomplete digraph $D$ and distinct vertices $x,y$, decides whether $D$ has a Hamiltonian $(x,y)$-path and constructs one whenever it exists.

Background

The paper identifies an extension of the prescribed-endpoint Hamiltonian-path problem from semicomplete digraphs to extended semicomplete and locally semicomplete digraphs. The conjecture asks for both polynomial-time recognition and construction.

The paper's results settle the connected nonstrong locally semicomplete case and several other tractable cases, but they do not resolve the full extended-sem​​icomplete formulation. In particular, the paper explicitly notes that the general extended-tournament version remains unresolved.

References

In the same survey, Bang-Jensen and Gutin proposed the following extension to two larger classes . There is a polynomial-time algorithm that, given an extended semicomplete or locally semicomplete digraph $D$ and distinct vertices $x,y\in V(D)$, decides whether $D$ has a Hamiltonian $(x,y)$-path and finds one whenever it exists.

— Paths with Prescribed Endpoints in Semicomplete and Locally Semicomplete Digraphs  (2608.17439 - Bai et al., 18 Aug 2026) in Conjecture 2 in Section 1