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 \({\mathcal O}^*(1.2051^n)\) time algorithm that solves it for this graph class. As a consequence, we can also solve this problem in \({\mathcal O}^*(1.2051^n)\) time for the classes of n-vertex P 6-free graphs and split graphs. This is an improvement upon a recent \({\mathcal O}^*(1.5790^n)\) 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.
This work has been supported by EPSRC (EP/D053633/1).
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Paulusma, D., van Rooij, J.M.M. (2009). On Partitioning a Graph into Two Connected Subgraphs. In: Dong, Y., Du, DZ., Ibarra, O. (eds) Algorithms and Computation. ISAAC 2009. Lecture Notes in Computer Science, vol 5878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10631-6_122
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DOI: https://doi.org/10.1007/978-3-642-10631-6_122
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