Connecting Terminals and 2-Disjoint Connected Subgraphs

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 \T | |T |−2 ) · 3 |V \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.