Polynomial kernels for tracking shortest paths

Abstract

Given an undirected graph G=(V,E), vertices s,t∈V, and an integer k, Tracking Shortest Paths requires deciding whether there exists a set of k vertices T⊆V such that for any two distinct shortest paths between s and t, say P1 and P2, we have T∩V(P1)≠T∩V(P2). In this paper, we give the first polynomial size kernel for the problem. Specifically we show the existence of a kernel with O(k2) vertices and edges in general graphs and a kernel with O(k) vertices and edges in planar graphs for the Tracking Paths in DAG problem. This problem admits a polynomial parameter transformation to Tracking Shortest Paths, and this implies a kernel with O(k4) vertices and edges for Tracking Shortest Paths in general graphs and a kernel with O(k2) vertices and edges in planar graphs. Based on the above we also give a single exponential algorithm for Tracking Shortest Paths in planar graphs.