頂点ラベル付きグラフの零距離同値類

A nonnegative vertex labeling of a connected graph defines a pseudoultrametric by minimizing the largest label along a path. Positive labels on at least one endpoint of each edge need not prevent distinct vertices from having distance zero. We characterize prescribed zero-distance classes by decreasing connected vertex sets. For finite classes this becomes connectivity of a finite auxiliary graph; for classes with countable complement it becomes connectivity after every finite deletion outside the class. The finite-class criterion gives a precise restricted version of a conjecture of Dovgoshey and Rovenska. The unrestricted conjecture is false: a labeling of the subdivided halved Farey graph collapses all its original vertices, although every pair has at most three internally disjoint paths. The bound three is optimal. We also realize infinite edge-connectivity classes by subdivision labelings, identify a compact metric quotient of the counterexample, and relate universal separation to the known criterion based on vertices that recur along a path sequence.