A Faster Parameterized Algorithm for Treedepth

TLDR: In this paper, an n-vertex graph, a tree decomposition of width w, and an integer t are used to decide whether the input graph has treedepth at most t in time 2 O(wt) ·n.

Abstract: The width measure treedepth, also known as vertex ranking, centered coloring and elimination tree height, is a well-established notion which has recently seen a resurgence of interest. We present an algorithm which—given as input an n-vertex graph, a tree decomposition of width w, and an integer t—decides whether the input graph has treedepth at most t in time 2 O(wt) ·n. We use this to construct further algorithms which do not require a tree decomposition as part of their input: A simple algorithm which decides treedepth in linear time for a fixed t, thus answering an open question posed by Ossona de Mendez and Nesetřil as to whether such an algorithm exists, a fast algorithm with running time \(2^{O(t^2)} \cdot n\) and an algorithm for chordal graphs with running time 2 O(t logt)·n.