A split graph is a graph whose vertices can be partitioned into a clique and an independent set. We study numerous problems on split graphs, namely the k-Vertex-Disjoint Paths, k-Cycle, k-Path and k-ℓ-Stable Set problems. In the k-Vertex-Disjoint Paths problem, we are given a graph 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-Cycle/k-Path problem, we are given a graph and are asked whether there is a path/cycle of length k. The k-ℓ-Stable Set problem takes a graph and an integer k as input, and asks whether the graph has a subset of k vertices such that the distance between every two vertices in the subset is at least ℓ+ 1. It is known that all the above problems are NP-complete on split graphs. We derive a 4k-vertex kernel for the k-Vertex-Disjoint Paths problem and an O (k 2)-vertex kernel for both the k-Path problem and the k-Cycle problem. Concerning the k-ℓ-Stable Set problem, for ℓ= 1 or ℓ≥ 3, the problem is polynomial-time solvable on split graphs. For ℓ= 2, we prove that the k-ℓ-Stable Set problem is W [1]-complete on split graphs, with respect to k. However, if the given split graph contains no K 1, r as an induced subgraph, and every vertex in the independent set of the split graph has degree at most d, we derive a linear vertex kernel for the k-2-Stable Set problem, where both r and d are constants.