Structural Parameterizations of Tracking Paths Problem

TL;DR

This paper investigates the Tracking Paths problem, proving its fixed-parameter tractability when parameterized by certain structural graph parameters, thus advancing understanding of its computational complexity.

Contribution

It introduces the first FPT algorithms for Tracking Paths based on structural parameters like vertex cover and cluster vertex deletion set.

Findings

FPT algorithms developed for vertex cover parameter

FPT algorithms developed for cluster vertex deletion set parameter

Advances understanding of the problem's complexity in structured graphs

Abstract

Given a graph $G$ with source and destination vertices $s,t\in V(G)$ respectively, \textsc{Tracking Paths} asks for a minimum set of vertices $T\subseteq V(G)$, such that the sequence of vertices encountered in each simple path from $s$ to $t$ is unique. The problem was proven \textsc{NP}-hard \cite{tr-j} and was found to admit a quadratic kernel when parameterized by the size of the desired solution \cite{quadratic}. Following recent trends, for the first time, we study \textsc{Tracking Paths} with respect to structural parameters of the input graph, parameters that measure how far the input graph is, from an easy instance. We prove that \textsc{Tracking Paths} admits fixed-parameter tractable (\textsc{FPT}) algorithms when parameterized by the size of vertex cover, and the size of cluster vertex deletion set for the input graph.