A polynomial bound on the pathwidth of graphs edge-coverable by $k$ shortest paths

2025-10-03

AI Summary

This paper addresses the open problem of bounding the pathwidth of graphs whose edge set can be covered by $k$ shortest paths—previously known only to admit an exponential upper bound of $O(3^k)$. We establish the first polynomial upper bound of $O(k^4)$, and prove that the bound is tight for $k leq 3$, where the pathwidth equals $k$. Furthermore, we show that graphs whose edges are covered by two isometric trees have treewidth at most $2$. Our approach integrates structural analysis of shortest paths, tree decomposition techniques, and constructive induction to quantify the relationship between edge coverings and pathwidth. This work reduces the best-known pathwidth bound from exponential to polynomial, revealing a deep connection between isometric tree coverings and classical width parameters (treewidth and pathwidth). It provides novel tools and benchmarks for parameterized structural graph theory.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• 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
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Dumas, Foucaud, Perez and Todinca (2024) recently proved that every graph whose edges can be covered by $k$ shortest paths has pathwidth at most $O(3^k)$. In this paper, we improve this upper bound on the pathwidth to a polynomial one; namely, we show that every graph whose edge set can be covered by $k$ shortest paths has pathwidth $O(k^4)$, answering a question from the same paper. Moreover, we prove that when $kleq 3$, every such graph has pathwidth at most $k$ (and this bound is tight). Finally, we show that even though there exist graphs with arbitrarily large treewidth whose vertex set can be covered by $2$ isometric trees, every graph whose set of edges can be covered by $2$ isometric trees has treewidth at most $2$.

Problem

• Improving pathwidth bound for graphs covered by k shortest paths
• Establishing polynomial pathwidth bound of O(k^4)
• Analyzing treewidth for graphs covered by isometric trees

Innovation

• Improved pathwidth bound to polynomial O(k^4)
• Tight pathwidth bound of k for k≤3
• Bounded treewidth for edge-cover by isometric trees