Fully decomposable split graphs

We discuss various questions around partitioning a split graph into connected parts. Our main result is a polynomial time algorithm that decides whether a given split graph is fully decomposable, i.e., whether it can be partitioned into connected parts of order a 1,a 2,...,a k for every a 1,a 2,...,a k summing up to the order of the graph. In contrast, we show that the decision problem whether a given split graph can be partitioned into connected parts of order a 1,a 2,...,a k for a given partition a 1,a 2,...,a k of the order of the graph, is NP-hard