This article proposes the adjacency method for generating all connected subsets of a graph without generating any disconnected subset. The motivation for the method is an algorithm for finding all candidate clusters of cancer cases in a specified geographic region. Tango and Takahashi (2005) suggested the ergodic method to find all candidate clusters for flexibly shaped spatial scan statistics. The adjacency method is an improvement of the ergodic method. A slight modification of the adjacency method can also be used to determine the connectivity of a graph. A recursive algorithm for finding all connected subsets is proposed in this article. Given a graph with n vertices, the cost of storage of the algorithm is O(n²) and the cost of the calculation is bounded by |Ω|* O(n²), where Ω is the collection of all connected subsets and |Ω| is the cardinality of Ω. In some situations, |Ω| is far less than 2ⁿ, the number of all subsets of the graph. For example, for a string with length n, , where δ > 0.