Trees With Path-Stable Center

Abstract

We study the notion of path-congruence \(\Phi: T_1 \rightarrow T_2\) between two trees \(T_1\) and \(T_2\). We introduce the concept of the trunk of a tree, and prove that, for any tree \(T\), the trunk and the periphery of \(T\) are stable. We then give conditions for which the center of \(T\) is stable. One such condition is that the central vertices have degree \(2\). Also, the center is stable when the diameter of \(T\) is less than \(8\).