On the size of minimal separators for treedepth decomposition

Abstract

The treedepth of a graph is a parameter that describes how it is close to a star graph. Recently, many parameterized algorithms have been developed for solving various problems on graphs with small treedepth. Since those algorithms are based on treedepth decompositions of input graphs, many solvers are developed to find optimal treedepth decompositions. Many practical solvers enumerate minimal separators to construct a treedepth decomposition in a top-down way. The bottleneck of these algorithms is often the enumeration of minimal separators. Therefore, to speed up the computation, it is essential to reduce the number of minimal separator candidates that are used for an optimal solution. We show that, to obtain an optimal treedepth decomposition, it is sufficient to consider only minimal separators of size at most 2tw, where tw denotes the treewidth of the input graph. We also show that this upper bound is tight up to a constant.