A counterexample to the dominating set conjecture

The metric polytope metn is the polyhedron associated with all semimetrics on n nodes and defined by the triangle inequalities xij − xik − xjk ≤ 0 and xij + xik + xjk ≤ 2 for all triples i, j, k of {1, . . . , n}. In 1992 Monique Laurent and Svatopluk Poljak conjectured that every fractional vertex of the metric polytope is adjacent to some integral vertex. The conjecture holds for n ≤ 8 and, in particular, for the 1 550 825 600 vertices of met8. While the overwhelming majority of the known vertices of met9 satisfy the conjecture, we exhibit a fractional vertex not adjacent to any integral vertex.