A Graph Width Perspective on Partially Ordered Hamiltonian Paths and Cycles II: Vertex and Edge Deletion Numbers

TL;DR

The paper analyzes the complexity of finding Hamiltonian paths and cycles under precedence constraints (POHPP and MinPOHPP) with respect to graph width parameters defined by vertex/edge deletions to reach target graph classes. It establishes $W[1]$-hardness for vertex distance to path and vertex distance to clique, while obtaining $XP$-time algorithms for vertex distance to outerplanar and to block, and $FPT$ algorithms for edge distance to block; it also reports para-$NP$-hardness for the edge clique cover number. The results delineate which deletion-based width parameters yield tractable POHPP variants and where intractability persists, guiding the design of practical algorithms under structural graph restrictions. The work complements prior studies on POHPP by mapping the boundary between hardness and tractability across a spectrum of width parameters, including twin-cover and modular decompositions.

Abstract

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.

Figures (1)

• Figure 1: Diagram illustrating the complexity results for (Min)POHPP for different graph width parameters. A directed solid edge from parameter $P$ to parameter $Q$ means that a bounded value of $P$ implies a bounded value for $Q$. A directed dashed edge implies that this relation does not hold in general but for traceable graphs, i.e., graphs having a Hamiltonian path. If a directed solid path from $P$ to $Q$ is missing, then parameter $Q$ is unbounded for the graphs of bounded $P$. The same holds for the traceable graphs if there is also no path using dashed edges.