# Tracking Paths in Planar Graphs

## TL;DR

This paper investigates the NP-complete problem of tracking paths in planar graphs, providing a 4-approximation algorithm and a linear-time solution for graphs with bounded clique width when a clique decomposition is available.

## Contribution

It proves NP-completeness for planar graphs and offers a 4-approximation algorithm, along with a linear-time solution for graphs with bounded clique width using Courcelle's theorem.

## Key findings

- NP-complete for planar graphs
- 4-approximation algorithm provided
- Linear-time solution for bounded clique width graphs

## Abstract

We consider the NP-complete problem of tracking paths in a graph, first introduced by Banik et. al. [3]. Given an undirected graph with a source $s$ and a destination $t$, find the smallest subset of vertices whose intersection with any $s-t$ path results in a unique sequence. In this paper, we show that this problem remains NP-complete when the graph is planar and we give a 4-approximation algorithm in this setting. We also show, via Courcelle's theorem, that it can be solved in linear time for graphs of bounded-clique width, when its clique decomposition is given in advance.