On the degrees of vertices in locally finite graphs which possess a certain edge deletion property (Q1068850)

A locally finite graph is said to be stable if its edge set is nonempty and if, for each edge e of G, there is a component of G-e which is isomorphic to G. Such is the infinite tree \(T_ n\) in which every vertex has degree \(n+1\) except for one vertex which has degree n. A graph is said to be bidegreed if its degree set contains only two elements. The author proves that the graphs \(T_ n\) are the only locally finite bidegreed stable graphs. The question of characterizing degree sets of locally finite stable graphs is raised and attributed to E. Harzheim. The author has some relevant conjectures.