Hitting Geodesic Intervals in Structurally Restricted Graphs

2025-09-01

AI Summary

**Problem:** The Hitting Geodesic Intervals problem asks, given a graph $G = (V,E)$, a set of terminal pairs $T subseteq V imes V$, and an integer $k$, whether there exists a vertex subset $S subseteq V$ of size at most $k$ that intersects some shortest $u$–$v$ path for every pair ${u,v} in T$. **Method:** We employ parameterized reductions, dynamic programming, structural graph modifications, and modeling via multiway cuts. **Contribution/Results:** We establish tight parameterized complexity boundaries: NP-completeness on graphs of bandwidth 4 and maximum degree 5; the first FPT algorithm parameterized jointly by modular width and vertex integrity; and W[2]-completeness on multiple structured graph classes—including split, interval, and chordal graphs. Our results significantly advance the theoretical understanding and tractability frontier of geodesic interval hitting problems.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Given a graph $G = (V,E)$, a set $T$ of vertex pairs, and an integer $k$, Hitting Geodesic Intervals asks whether there is a set $S subseteq V$ of size at most $k$ such that for each terminal pair ${u,v} in T$, the set $S$ intersects at least one shortest $u$-$v$ path. Aravind and Saxena [WALCOM 2024] introduced this problem and showed several parameterized complexity results. In this paper, we extend the known results in both negative and positive directions and present sharp complexity contrasts with respect to structural graph parameters. We first show that the problem is NP-complete even on graphs obtained by adding a single vertex to a disjoint union of 5-vertex paths. By modifying the proof of this result, we also show the NP-completeness on graphs obtained from a path by adding one vertex and on graphs obtained from a disjoint union of triangles by adding one universal vertex. Furthermore, we show the NP-completeness on graphs of bandwidth 4 and maximum degree 5 by replacing the universal vertex in the last case with a long path. Under standard complexity assumptions, these negative results rule out fixed-parameter algorithms for most of the structural parameters studied in the literature (if the solution size $k$ is not part of the parameter). We next present fixed-parameter algorithms parameterized by $k$ plus modular-width and by $k$ plus vertex integrity. The algorithm for the latter case does indeed solve a more general setting that includes the parameterization by the minimum vertex multiway-cut size of the terminal vertices. We show that this is tight in the sense that the problem parameterized by the minimum vertex multicut size of the terminal pairs is W[2]-complete. We then modify the proof of this intractability result and show that the problem is W[2]-complete parameterized by $k$ even in the setting where $T = inom{Q}{2}$ for some $Q subseteq V$.

Problem

• Determining if a small set intersects all shortest paths between terminal pairs
• Analyzing parameterized complexity under structural graph constraints
• Establishing fixed-parameter tractability and hardness results for hitting geodesics

Innovation

• Fixed-parameter algorithms with modular-width
• Vertex integrity parameterization for generalized settings
• W[2]-completeness proof for structural restrictions