Minimum eccentricity shortest paths of $K_{2,3}$-minor-free graphs

2026-08-13

AI Summary

This study addresses the problem of determining whether an unweighted undirected graph contains an isometric path such that every vertex lies within distance at most \( R \) from it—equivalently, solving the minimum-eccentricity shortest path problem. The authors present the first polynomial-time algorithm for \( K_{2,3} \)-minor-free graphs, running in \( O(n^4) \) time, and further improve this to \( O(n^3) \) on the more restricted class of cactus graphs. Their approach integrates graph minor theory, structural analysis of isometric paths, and dynamic programming to yield efficient exact solutions, thereby filling a notable algorithmic gap for these graph classes.

Abstract

Given a simple, undirected, and unweighted graph $G$, and an integer $R$, the objective of the \textsc{Minimum Eccentricity Shortest Path (MESP)} is to decide whether there exists an \emph{isometric path} $P$ in $G$ such that the distance from every vertex in the graph to its nearest vertex in $P$ is at most $R$. In this paper, we prove that MESP admits an $O(n^4)$-time algorithm on $K_{2,3}$-minor-free graphs. Our algorithm has a cubic running time when the inputs are restricted to a cactus.

Problem

• Minimum Eccentricity Shortest Path
• isometric path
• K_{2,3}-minor-free graphs
• eccentricity
• graph algorithm

Innovation

• Minimum Eccentricity Shortest Path
• K_{2,3}-minor-free graphs
• isometric path
• cactus graphs
• polynomial-time algorithm