We consider the problem of finding a Hamiltonian path or cycle with precedence constraints in the form of a partial order on the vertex set. We study the complexity for graph width parameters for which the ordinary problems $\mathsf{Hamiltonian\ Path}$ and $\mathsf{Hamiltonian\ Cycle}$ are in $\mathsf{FPT}$. In particular, we focus on parameters that describe how many vertices and edges have to be deleted to become a member of a certain graph class. We show that the problems are $\mathsf{W[1]}$-hard for such restricted cases as vertex distance to path and vertex distance to clique. We complement these results by showing that the problems can be solved in $\mathsf{XP}$ time for vertex distance to outerplanar and vertex distance to block. Furthermore, we present some $\mathsf{FPT}$ algorithms, e.g., for edge distance to block. Additionally, we prove para-$\mathsf{NP}$-hardness when considered with the edge clique cover number.

翻译：我们研究在顶点集上存在偏序形式优先约束条件下寻找哈密顿路径或哈密顿环的问题。针对普通问题 $\mathsf{Hamiltonian\ Path}$ 和 $\mathsf{Hamiltonian\ Cycle}$ 属于 $\mathsf{FPT}$ 类的图宽度参数，我们考察了该约束问题的计算复杂度。特别地，我们聚焦于描述需要删除多少顶点和边才能使图成为特定图类成员的参数。我们证明，即使在顶点距离至路径和顶点距离至团等受限情况下，这些问题也是 $\mathsf{W[1]}$-难的。作为补充，我们证明了对于顶点距离至外平面图和顶点距离至块图的情况，这些问题可在 $\mathsf{XP}$ 时间内求解。此外，我们提出了一些 $\mathsf{FPT}$ 算法，例如针对边距离至块图的情况。另外，我们证明了当结合边团覆盖数考虑时，该问题具有 para-$\mathsf{NP}$-难性。