A Graph Width Perspective on Partially Ordered Hamiltonian Paths

We consider the problem of finding a Hamiltonian path with precedence constraints in the form of a partial order on the vertex set. This problem is known as Partially Ordered Hamiltonian Path Problem (POHPP). Here, we study the complexity for graph width parameters for which the ordinary Hamiltonian Path problem is in $\mathsf{FPT}$. We show that POHPP is $\mathsf{NP}$-complete for graphs of pathwidth 4. We complement this result by giving polynomial-time algorithms for graphs of pathwidth 3 and treewidth 2. Furthermore, we show that POHPP is $\mathsf{NP}$-hard for graphs of clique cover number 2 and $\mathsf{W[1]}$-hard for some distance-to-$\mathcal{G}$ parameters, including distance to path and distance to clique. In addition, we present $\mathsf{XP}$ and $\mathsf{FPT}$ algorithms for parameters such as distance to block and feedback edge set number.