Geodetic Set on Graphs of Constant Pathwidth and Feedback Vertex Set Number

Abstract

In the Geodetic Set problem, the input consists of a graph G and a positive integer k. The goal is to determine whether there exists a subset S of vertices of size k such that every vertex in the graph is included in a shortest path between two vertices in S. Kellerhals and Koana [IPEC 2020; J. Graph Algorithms Appl 2022] proved that the problem is W[1]-hard when parameterized by the pathwidth or the feedback vertex set number of the input graph. They posed the question of whether the problem admits an XP-algorithm when parameterized by the combination of these two parameters. We answer this in the negative by proving that the problem remains NP-hard even on graphs of constant pathwidth and feedback vertex set number.