On partitioning a graph into two connected subgraphs

Abstract

Suppose a graph G is given with two vertex-disjoint sets of vertices Z 1 and Z 2. Can we partition the remaining vertices of G such that we obtain two connected vertex-disjoint subgraphs of G that contain Z 1 and Z 2, 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 Z 1 and Z 2 each contain a connected set that dominates all vertices in V\(Z 1∪Z 2). We present an time algorithm that solves it for this graph class. As a consequence, we can also solve this problem in time for the classes of n-vertex P 6-free graphs and split graphs. This is an improvement upon a recent 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. © 2009 Springer-Verlag Berlin Heidelberg.