Kernelization of Two Path Searching Problems on Split Graphs

In the k-Vertex-Disjoint Paths problem, we are given a graph G and k terminal pairs of vertices, and are asked whether there is a set of k vertex-disjoint paths linking these terminal pairs, respectively. In the k-Path problem, we are given a graph and are asked whether there is a path of length k. It is known that both problems are NP-hard even in split graphs, which are the graphs whose vertices can be partitioned into a clique and an independent set. We study kernelization for the two problems in split graphs. In particular, we derive a 4k vertex-kernel for the k-Vertex-Disjoint Paths problem and a 3/2k(2) + 1/2k vertex-kernel for the k-Path problem.