The median and centroid of an arbitrary graph G are two different generalizations of the branch weight centroid of a tree. As such, they are closely related, but they can actually be disjoint. On the one hand, they are, for example, always contained in the same block of any connected graph G. However, they can be arbitrarily far apart. Specifically, given any three graphs H, J, and K, and a positive integer k ≥ 4, there exists a graph G with center, median, and centroid subgraphs isomorphic to H, J, and K, respectively, and the distance between any two of these subgraphs is at least k.
