W[2]-completeness of Min-Internal for clique-starter searches beyond Generic Search

Determine whether, for clique-starter graph searches other than Generic Search (e.g., Breadth-First Search, Lexicographic Breadth-First Search, or related clique-starter searches), the Min-Internal first-in tree problem parameterized by the number of internal vertices k is W[2]-complete (rather than merely W[2]-hard).

Background

The authors prove that Min-Internal for first-in trees is W[2]-hard (and NP-hard) on split graphs for any clique-starter search. For Generic Search specifically, Min-Internal coincides with Connected Dominating Set, which is known to be W[2]-complete, yielding completeness for that search.

They explicitly state that, beyond Generic Search, it remains unresolved whether the W[2]-hardness can be strengthened to W[2]-completeness for other clique-starter searches.

References

Note that we leave open whether there are clique starters other than GS for which Min-Internal is not just W[2]-hard but also W[2]-complete.

— Breadth-First Search Trees with Many or Few Leaves  (2604.00691 - Beisegel et al., 1 Apr 2026) in Section 4.1 (Hardness for Search Trees with Few Internal Vertices), paragraph after Theorem 4.2