In the theory of accessible categories, pure subobjects, i.e. filtered colimits of split monomorphisms, play an important role. Here we investigate pure quotients, i.e., filtered colimits of split epimorphisms. For example, in abelian, finitely accessible categories, these are precisely the cokernels of pure subobjects, and pure subobjects are precisely the kernels of pure quotients.
Let $G$ be a connected, undirected graph without loops and without multiple edges. For a pair of distinct vertices $u$ and $v$, a minimum $\{u,v\}$-separating set is a smallest set of edges in $G$ whose removal disconnects $u$ and $v$. The edge connectivity of $G$, denoted $\lambda (G)$, is defined to be the minimum cardinality of a minimum $\{u,v\}$-separating set as $u$ and $v$ range over all pairs of distinct vertices in $G$. We introduce and investigate the eavesdropping number, denoted $\varepsilon (G)$, which is defined to be the maximum cardinality of a minimum $\{u,v\}$-separating set as $u$ and $v$ range over all pairs of distinct vertices in $G$. Results are presented for regular graphs and maximally locally connected graphs, as well as for a number of common families of graphs.