On Partitioning a Graph into Two Connected Subgraphs

  • Authors:

  • Daniël Paulusma;Johan M. Rooij

  • Affiliations:

  • Department of Computer Science, University of Durham, Durham, England DH1 3LE;Department of Information and Computing Sciences, Universiteit Utrecht, Utrecht, The Netherlands 3508TB

  • Venue:
  • ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
  • Year:
  • 2009

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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 驴 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.