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, that is, whether it can be partitioned into connected parts of orders α 1 , α 2 , … , α k for every α 1 , α 2 , … , α 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 orders α 1 , α 2 , … , α k for a given partition α 1 , α 2 , … , α k of the order of the graph, is NP-hard.