Parameterized complexity of Max-Leaf search trees

Determine the parameterized complexity of Max-Leaf search trees for graph searches other than Depth-First Search and Lexicographic Depth-First Search, and develop a fixed-parameter algorithm for Max-Leaf trees of Depth-First Search and Lexicographic Depth-First Search that complements the known hardness results.

Background

The paper studies minimizing and maximizing the number of leaves in last-in search trees generated by graph searches. It establishes hardness results for Lexicographic Depth-First Search by transferring results from Depth-First Search, but the corresponding landscape for other searches remains unresolved. The authors specifically identify the maximizing variant, Max-Leaf, as incompletely understood and note the absence of fixed-parameter algorithms for both Depth-First Search and Lexicographic Depth-First Search.

References

In particular, the parameterized complexity of Max-Leaf is still largely unknown for all searches apart from (L)DFS, while for these two, no -algorithm is known to complement the -hardness~results.

— On Kernels and Leaves: Searching for Bare and Lush Trees  (2609.29451 - Beisegel et al., 24 Sep 2026) in Section Further Research