# On the status sequences of trees

## TL;DR

This paper explores the properties of status sequences in trees, establishing computational complexity results, characterizing certain classes of trees, and relating status sequences to graph partitions.

## Contribution

It proves NP-completeness of recognizing status sequences, characterizes status injective trees, and links status sequences to orbit and equitable partitions.

## Key findings

- Deciding if a tree has a given status sequence is NP-complete.
- Status injective trees are uniquely determined among trees.
- Relations between status sequences and graph partitions are established.

## Abstract

The status of a vertex $v$ in a connected graph is the sum of the distances from $v$ to all other vertices. The status sequence of a connected graph is the list of the statuses of all the vertices of the graph. In this paper we investigate the status sequences of trees. Particularly, we show that it is NP-complete to decide whether there exists a tree that has a given sequence of integers as its status sequence. We also present some results about trees whose status sequences are comprised of a few distinct numbers or many distinct numbers. In this direction, we provide a partial answer to a conjecture of Shang and Lin from 2011, showing that any status injective tree is unique among trees. Finally, we investigate how orbit partitions and equitable partitions relate to the status sequence.