AbstractIn this paper we consider a cutting process for random mappings. Specifically, for 0 <m<n, we consider the initial (uniform) random mapping digraphGnonnlabeled vertices, and we delete (if possible), uniformly and at random,mnoncyclic directed edges fromGn. The maximal random digraph consisting of the unicyclic components obtained after cutting themedges is called the trimmed random mapping and is denoted byG. If the number of noncyclic directed edges is less thanm, thenGconsists of the cycles, including loops, of the initial mappingGn. We consider the component structure of the trimmed mappingG. In particular, using the exact distribution we determine the asymptotic distribution of the size of a typical random connected component ofGasn,m→∞. This asymptotic distribution depends on the relationship betweennandmand we show that there are three distinct cases: (i)$m=o(\sqrt{n})$, (ii)$m=\beta\sqrt{n}$, whereβ> 0 is a fixed parameter, and (iii)$\sqrt{n}=o(m)$. This allows us to study the joint distribution of the order statistics of the normalized component sizes ofG. When$m=o(\sqrt{n})$, we obtain thePoisson–Dirichlet(1/2) distribution in the limit, whereas when$\sqrt{n}=o(m)$the limiting distribution is Poisson–Dirichlet(1). Convergence to the Poisson–Dirichlet(θ) distribution breaks down when$m=O(\sqrt{n})$, and in particular, there is no smooth transition from the${\cal P}$D(1/2) distribution to the${\cal P}$D(1) via the Poisson–Dirichlet distribution as the number of edges cut increases relative ton, the number of vertices inGn. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 30, 287–306, 2007