Partitions and Coverings of Trees by Bounded-Degree Subtrees

TL;DR

This paper investigates the minimum number of bounded-degree subtrees needed to partition or cover a tree's edges, providing explicit formulas, algorithms, and bounds related to tree structure and pathwidth.

Contribution

It offers an explicit formula and efficient algorithms for partitioning and covering trees with bounded-degree subtrees, and establishes bounds involving pathwidth.

Findings

Explicit formula for minimum number of degree-d subtrees in a partition.

Linear time algorithm for finding such partitions.

Polynomial time algorithm for minimum coverings and bounds related to pathwidth.

Abstract

This paper addresses the following questions for a given tree $T$ and integer $d\geq2$: (1) What is the minimum number of degree-$d$ subtrees that partition $E(T)$? (2) What is the minimum number of degree-$d$ subtrees that cover $E(T)$? We answer the first question by providing an explicit formula for the minimum number of subtrees, and we describe a linear time algorithm that finds the corresponding partition. For the second question, we present a polynomial time algorithm that computes a minimum covering. We then establish a tight bound on the number of subtrees in coverings of trees with given maximum degree and pathwidth. Our results show that pathwidth is the right parameter to consider when studying coverings of trees by degree-3 subtrees. We briefly consider coverings of general graphs by connected subgraphs of bounded degree.