Finite separating sets in locally finite graphs (Q1321991)

An \((m,n)\)-separator of an infinite graph \(\Gamma\) is a set of minimum cardinality \(\kappa_{m,n}\) whose deletion leaves at least \(m\) finite components and at least \(n\) infinite components. It is shown that a vertex of \(\Gamma\) of finite valence belongs to only finitely many \((0,2)\)-separators. Various results concerning the interrelation of \((m,n)\)-separators (especially (0,2)-separators) are obtained for locally finite graphs. Particular attention is given in the case that \(\Gamma\) is vertex-transitive or edge-transitive. For such \(\Gamma\) it is shown that \(\kappa_{0,2}< \kappa_{1,2}\), if \(0< \kappa_{1,2}<\infty\).