On the Size of Minimal Separators for Treedepth Decomposition

Treedepth decomposition has several practical applications and can be used to speed up many parameterized algorithms. There are several works aiming to design a scalable algorithm to compute exact treedepth decompositions. Those include works based on a set of all minimal separators. In those algorithms, although a number of minimal separators are enumerated, the minimal separators that are used for an optimal solution are empirically very small. Therefore, analyzing the upper bound on the size of minimal separators is an important problem because it has the potential to significantly reduce the computation time. A minimal separator $S$ is called an optimal top separator if $td(G) = |S| + td(G \backslash S)$, where $td(G)$ denotes the treedepth of $G$. Then, we have two theoretical results on the size of optimal top separators. (1) For any $G$, there is an optimal top separator $S$ such that $|S| \le 2tw(G)$, where $tw(G)$ is the treewidth of $G$. (2) For any $c < 2$, there exists a graph $G$ such that any optimal top separator $S$ of $G$ have $|S| > c \cdot tw(G)$, i.e., the first result gives a tight bound on the size of an optimal top separator.