Pathfinding in Self-Deleting Graphs

2025-07-16

AI Summary

This paper studies the path-finding problem in self-deleting graphs: given source $s$ and target $t$, each vertex $v$ is associated with a deletion function $f(v)$ specifying edges permanently removed upon visiting $v$; the goal is to find a shortest $s$–$t$ path under this dynamic edge-deletion constraint. Methodologically, the work integrates dynamic graph modeling, tree decompositions, feedback vertex set analysis, and tight parameterized reductions. Contributions include: (i) establishing the first systematic parameterized complexity landscape—proving W[1]-completeness parameterized by either the maximum outdegree of $f$ or the size of a minimum feedback vertex set; (ii) showing NP-hardness even on graphs of bounded treewidth and bandwidth; (iii) identifying FPT tractability boundaries—yielding FPT algorithms when $|f(v)|$ is bounded and the graph has bounded treewidth or feedback vertex number; and (iv) proving that the problem admits no polynomial kernel unless $ ext{NP} subseteq ext{coNP/poly}$.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this paper, we study the problem of pathfinding on traversal-dependent graphs, i.e., graphs whose edges change depending on the previously visited vertices. In particular, we study emph{self-deleting graphs}, introduced by Carmesin et al. (Sarah Carmesin, David Woller, David Parker, Miroslav Kulich, and Masoumeh Mansouri. The Hamiltonian cycle and travelling salesperson problems with traversal-dependent edge deletion. J. Comput. Sci.), which consist of a graph $G=(V, E)$ and a function $fcolon V ightarrow 2^E$, where $f(v)$ is the set of edges that will be deleted after visiting the vertex $v$. In the extsc{(Shortest) Self-Deleting $s$-$t$-path} problem we are given a self-deleting graph and its vertices $s$ and $t$, and we are asked to find a (shortest) path from $s$ to $t$, such that it does not traverse an edge in $f(v)$ after visiting $v$ for any vertex $v$. We prove that extsc{Self-Deleting $s$-$t$-path} is NP-hard even if the given graph is outerplanar, bipartite, has maximum degree $3$, bandwidth $2$ and $|f(v)|leq 1$ for each vertex $v$. We show that extsc{Shortest Self-Deleting $s$-$t$-path} is W[1]-complete parameterized by the length of the sought path and that extsc{Self-Deleting $s$-$t$-path} is W{1}-complete parameterized by the vertex cover number, feedback vertex set number and treedepth. We also show that the problem becomes FPT when we parameterize by the maximum size of $f(v)$ and several structural parameters. Lastly, we show that the problem does not admit a polynomial kernel even for parameterization by the vertex cover number and the maximum size of $f(v)$ combined already on 2-outerplanar graphs.

Problem

• Pathfinding in graphs with traversal-dependent edge deletions
• NP-hardness of Self-Deleting s-t-path in constrained graphs
• Parameterized complexity of Shortest Self-Deleting s-t-path

Innovation

• Pathfinding in traversal-dependent self-deleting graphs
• NP-hard proof for outerplanar bipartite graphs
• Parameterized complexity analysis for pathfinding