Some Structural Properties of a Least Central Subtree of a Tree

We consider the graph center problem in the joinsemilattice L(T ) of all subtrees of a tree T . A subtree S of a tree T is a central subtree of T if S has the minimum eccentricity in the joinsemilattice. The graph center of the joinsemilattice is the set of all central subtrees. A central subtree with the minimum number of points is a least central subtree of a tree T . Thus least central subtrees of T are, in some sense, the best possible connected substructures of T among all connected substructures. We show that every tree is a unique least central subtree of some larger tree. Our main result points out the importance of the cardinality of the nodes of degree two. Low cardinality guarantees uniqueness and explicit construction for the least central subtree.