Semi-stable and stable cacti

Abstract

Holton (1973) introduced the following concept. A graph G is semi-stable if there exists a point v in G for which Γ(Gv) = Γ(G)v: where Γ(G) is the automorphism group of G, Gv is the graph G with v deleted and Γ(G)v is the subgroup of Γ(G) that fixes v. We say G is semi-stable at v. A partial stabilising sequence in Gs1, v2,…vk of its points such that Γ(G)v1v2…vi ═ Γ(Gv1v2…v1) for i = 1,2,…,k. If there exists a partial stabilising sequence in G for which k equals the number of points of G then G is said to be stable (Holton (1973a)). Most natation and terminology in what follows is explained in Harary (1969).