Central Forests in Trees

TL;DR

This paper introduces central forests, a new family of structures in trees that generalize existing centrality concepts, along with an efficient algorithm for their construction and analysis of their properties.

Contribution

The paper defines central forests as a new two-parameter family of central structures in trees, providing an $O(n(m+k))$ algorithm for their construction and exploring their elementary properties.

Findings

Introduced the concept of central forests in trees.

Developed an efficient algorithm for constructing central forests.

Connected central forests to existing centrality problems.

Abstract

A new 2-parameter family of central structures in trees, called central forests, is introduced. Minieka's $m$-center problem and McMorris's and Reid's central-$k$-tree can be seen as special cases of central forests in trees. A central forest is defined as a forest $F$ of $m$ subtrees of a tree $T$, where each subtree has $k$ nodes, which minimizes the maximum distance between nodes not in $F$ and those in $F$. An $O(n(m+k))$ algorithm to construct such a central forest in trees is presented, where $n$ is the number of nodes in the tree. The algorithm either returns with a central forest, or with the largest $k$ for which a central forest of $m$ subtrees is possible. Some of the elementary properties of central forests are also studied.