On different "middle parts" of a tree

TL;DR

This paper investigates the maximum distances among key central parts of trees, such as the center, centroid, and subtree core, under various constraints like order, degree, and diameter.

Contribution

It provides new results on the maximum distances among tree centers and introduces a partial characterization for extremal structures related to subtree cores.

Findings

Maximum distances between center, centroid, and subtree core are determined for trees with fixed order.

Results extend to trees with given maximum degree and diameter.

Partial characterization of extremal trees for the subtree core problem is provided.

Abstract

We determine the maximum distance between any two of the center, centroid, and subtree core among trees with a given order. Corresponding results are obtained for trees with given maximum degree and also for trees with given diameter. The problem of the maximum distance between the centroid and the subtree core among trees with given order and diameter becomes difficult. It can be solved in terms of the problem of minimizing the number of root-containing subtrees in a rooted tree of given order and height. While the latter problem remains unsolved, we provide a partial characterization of the extremal structure.