Abstract
Given a graph G = (V,E) and a set of terminal vertices T we say that a superset S of T is T-connecting if S induces a connected graph, and S is minimal if no strict subset of S is T-connecting. In this paper we prove that there are at most \({|V \setminus T| \choose |T|-2} \cdot 3^{\frac{|V \setminus T|}{3}}\) minimal T-connecting sets when |T| ≤ n/3 and that these can be enumerated within a polynomial factor of this bound. This generalizes the algorithm for enumerating all induced paths between a pair of vertices, corresponding to the case |T| = 2. We apply our enumeration algorithm to solve the 2-Disjoint Connected Subgraphs problem in time O *(1.7804n), improving on the recent O *(1.933n) algorithm of Cygan et al. 2012 LATIN paper.
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Telle, J.A., Villanger, Y. (2013). Connecting Terminals and 2-Disjoint Connected Subgraphs. In: Brandstädt, A., Jansen, K., Reischuk, R. (eds) Graph-Theoretic Concepts in Computer Science. WG 2013. Lecture Notes in Computer Science, vol 8165. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45043-3_36
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DOI: https://doi.org/10.1007/978-3-642-45043-3_36
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