Primitivity and ends of graphs (Q1343171)

A graph \(\Gamma\) is said to be primitive if its automorphism group \(\Aut(\Gamma)\) acts on its vertex set as a primitive group. A locally finite graph \(\Gamma\) is said to be accessible if there exists a number \(k\) such that any two ends of \(\Gamma\) can be separated by the removal of at most \(k\) edges. Theorem: Every locally finite, primitive graph is accessible. The author then proves that every connected, locally finite, primitive graph \(\Gamma\) has the following interesting property that renders transparent its end structure: there always exists a pair of vertices \(a\) and \(b\) in \(\Gamma\) such that the graph \(\Gamma'\) on the same vertex set as \(\Gamma\) but whose edge set is the orbit under \(\Aut(\Gamma)\) of \(\{a,b\}\) has connectivity 1 and each lobe (or block) of \(\Gamma'\) has at most one end.