A Parameterized Complexity Analysis of Bounded Height Depth-first Search Trees

Computing bounded depth decompositions is a bottleneck in many applications of the treedepth parameter. The fastest known algorithm, which is due to Reidl, Rossmanith, Sánchez Villaamil, and Sikdar [ICALP 2014], runs in $2^{\mathcal{O}(k^2)}\cdot n$ time and it is a big open problem whether the dependency on $k$ can be improved to $2^{o(k^2)}\cdot n^{\mathcal{O}(1)}$. We show that the related problem of finding DFS trees of bounded height can be solved faster in $2^{\mathcal{O}(k \log k)}\cdot n$ time. As DFS trees are treedepth decompositions, this circumvents the above mentioned bottleneck for this subclass of graphs of bounded treedepth. This problem has recently found attention independently under the name Minimum Height Lineal Topology (MinHLT) and our algorithm gives a positive answer to an open problem posed by Golovach [Dagstuhl Reports, 2023]. We complement our main result by studying the complexity of MinHLT and related problems in several other settings. First, we show that it remains NP-complete on chordal graphs, and give an FPT-algorithm on chordal graphs for the dual problem, asking for a DFS tree of height at most $n-k$, parameterized by $k$. The parameterized complexity of Dual MinHLT on general graphs is wide open. Lastly, we show that Dual MinHLT and two other problems concerned with finding DFS trees with few or many leaves are FPT parameterized by $k$ plus the treewidth of the input graph.