A Direct Proof of a Theorem on Detachments of Finite Graphs

Let \(G\) be a finite graph with vertices \(\xi_1, \ldots, \xi_n\), and let \(S_1, \ldots, S_n\) be disjoint non-empty finite sets. We give a new proof of a theorem characterizing the least possible number of connected components of a graph \(D\) such that \(V(D) = S_1 \cup \cdots \cup S_n\), \(E(D) = E(G)\) and, when an edge \(\lambda\) joins vertices \(\xi_i, \xi_j\) in \(G\), it is required to join some element of \(S_i\) to some element of \(S_j\) in \(D\) (so that, informally, \(D\) arises from \(G\) by splitting each vertex \(\xi_i\) into \(|S_i|\) vertices).