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 $\alpha_1,\alpha_2,\dots, \alpha_k$ for every $\alpha_1,\alpha_2,\ldots,\alpha_k$ 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 $\alpha_1,\alpha_2,\ldots,\alpha_k$ for a given partition $\alpha_1,\alpha_2,\ldots,\alpha_k$ of the order of the graph, is NP-hard