On partitioning a graph into two connected subgraphs

Suppose a graph G is given with two vertex-disjoint sets of vertices Z₁ and Z₂. Can we partition the remaining vertices of G such that we obtain two connected vertex-disjoint subgraphs of G that contain Z₁ and Z₂, respectively? This problem is known as the 2-Disjoint Connected Subgraphs problem. It is already NP-complete for the class of n-vertex graphs G=(V,E) in which Z₁ and Z₂ each contain a connected set that dominates all vertices in V?(Z₁∪Z₂). We present an O^∗(1.2051ⁿ) time algorithm that solves it for this graph class. As a consequence, we can also solve this problem in O^∗(1.2051ⁿ) time for the classes of n-vertex P₆-free graphs and split graphs. This is an improvement upon a recent O^∗(1.5790ⁿ) time algorithm for these two classes. Our approach translates the problem to a generalized version of hypergraph 2-coloring and combines inclusion/exclusion with measure and conquer.