Tracking Paths in Planar 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.