Clustering is an important data exploration task in chance discovery as well as in data mining. The first hierarchical clustering dates back to 1951 by K. Florek; since then, there have been numerous algorithms. However, there is no consensus among researchers as to what constitutes a cluster; the choice of the cluster is application-dependent. Although clustering is sometimes evaluated by interpretability of clusters, few studies have been done to reveal the interpretation aspect of clusters. This paper explains development of a new clustering algorithm by graph-based partitioning which aims to simplify interpretation of clusters. Two typical cluster types are considered: a star and a diamond. A star is a cluster with explicit shared context, represented by a central node. A diamond is a cluster with shared context, whose main cause of the context is implicit and hidden. These two types are very easy to understand. We elicit these types of clusters from a given weighted linkage graph. Minimization of weight of the graph cut is also considered. We show some examples and explain the effectiveness of our method.