Polynomial kernels for tracking shortest paths

Given an undirected graph G = (V, E), vertices s, t E V, and an integer k, TRACKING SHORTEST PATHS requires deciding whether there exists a set of k vertices T c_ V such that for any two distinct shortest paths between s and t, say P1 and P2, we have T n V(P1) =? T n 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. (c) 2022 Elsevier B.V. All rights reserved.