FPT status of Max-Internal for clique-starter searches beyond Generic Search

Ascertain whether the Max-Internal first-in tree problem, parameterized by the number of internal vertices k, is fixed-parameter tractable for clique-starter searches other than Generic Search.

Background

The paper shows W[1]-hardness (and NP-hardness) of Max-Internal for first-in trees across clique-starter searches and proves W[1]-completeness for Generic Search. For BFS, the authors provide an XP-time algorithm but do not establish FPT membership.

They explicitly note that, for clique-starter searches other than Generic Search, it remains open whether Max-Internal is in FPT.

References

Again, we have to leave open whether Max-Internal of other clique starters is in FPT.

— Breadth-First Search Trees with Many or Few Leaves  (2604.00691 - Beisegel et al., 1 Apr 2026) in Section 4.2 (Hardness for Search Trees with Many Internal Vertices), paragraph following Theorem 4.5