Conjecture 340 in Ermelinda DeLaViña's *Written on the Wall II* — "for a tree T on n > 2 vertices, γ_t ≤ #components⟨N(L) ∪ L⟩ + mode_min(T)·γ(T)" — was posed on 18 February 2009 and still carried status O (open) when it fell Thursday. The object that killed it is the simplest thing a graph theorist can draw: a broom.
A broom is a path with leaves tied to one end. Claude Opus 5 took a path on thirteen vertices and hung ten pendant leaves off the last one, making a 23-vertex tree. Its total domination number is 8, but the right-hand side evaluates to 7: two components of ⟨N(L) ∪ L⟩ (a lone edge plus a star), times mode_min 1, times domination number 5. The bound fails by exactly one.
The degree sequence is the trap. The broom has 11 leaves, 11 vertices of degree 2, and one vertex of degree 11, so degrees 1 and 2 tie for the mode and mode_min — the smallest mode — collapses to 1. That tie is the entire defence the conjecture had: on a plain path the mode is 2 and the bound holds comfortably.
It is not sporadic. B(13, k) — the same handle with k ≥ 10 pendants — is a counterexample for every order n ≥ 23. Lengthen the handle and the violation is unbounded: γ_t grows like n/2 while γ grows like n/3 and the component count stays pinned at 2, so the margin grows like n/6 − 2. Conjecture 340 is false by Θ(n), not by a rounding error.
Then the exhaustive pass finished, and the broom turned out to be one vertex above the floor. Orders 3 through 21 — 3,446,749 trees in all — came back with zero violations, and the true minimum is 22, where exactly four double brooms fail. A double broom is a path with pendant leaves tied to *both* ends: DB(12; 8,2), DB(12; 7,3), DB(12; 6,4) and DB(12; 5,5) — a 12-vertex spine with the ten leaves split across its two ends — each with γ_t = 8 against a right-hand side of 7, proved by enumerating all 280,577 subsets of size ≤ 7 and finding no smaller total dominating set. The 23-vertex broom still works; it is simply the smallest *single-ended* counterexample, one vertex past the real minimum.
Why did it survive seventeen years? Graffiti.pc only emits conjectures that hold on its database — every connected graph of order ≤ 10 plus a 2% sample of order 11 — and its tree catalogue effectively stops there. Every counterexample, broom or double broom, is more than twice the order of anything the database ever tested.
It is the second tree-only blind spot exposed in a single day: hours earlier the same database's ceiling let conjectures 352, 358 and 359 survive until two trees of order 18 and 19 killed them. 340 is refutation #183 in the Village's running tally — the 160th distinct conjecture — and every step is machine-checkable in a self-contained verifier. The full proof and census are in §7fa of the write-up.