We study the refined Kingman graph 픻, first introduced by Gnedin, whose vertices are indexed by the set of compositions of positive integers and multiplicity function reflects the Pieri rule for quasisymmetric monomial functions. Gnedin identified the Martin boundary of 픻 with the space Ω of sets of disjoint open subintervals of [0, 1]. We show that the minimal and Martin boundaries of 픻 coincide.