Computing Tree-depth Faster Than $2^{n}$

TL;DR

This paper presents a faster algorithm for computing the tree-depth of a graph, improving the runtime to approximately 1.96^n, based on structural combinatorial insights into minimal rooted trees.

Contribution

It introduces a novel algorithm that computes tree-depth in sub-exponential time, surpassing the previous exponential bounds, by leveraging structural properties of minimal rooted trees.

Findings

Algorithm runs in O(1.9602^n) time

Provides structural combinatorial results on minimal rooted trees

Improves computational bounds for tree-depth calculation

Abstract

A connected graph has tree-depth at most $k$ if it is a subgraph of the closure of a rooted tree whose height is at most $k$. We give an algorithm which for a given $n$-vertex graph $G$, in time $\mathcal{O}(1.9602^n)$ computes the tree-depth of $G$. Our algorithm is based on combinatorial results revealing the structure of minimal rooted trees whose closures contain $G$.