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

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.