If G is a graph with vertex set V(G) and (vertex) automorphism group γ(G), then a sequence s={vπ(i)} i=1 k of distinct vertices of G is a partial stabilising sequence for G if \(\Gamma \left( {G_{S_n } } \right) = \Gamma \left( G \right)_{S_n } \)for n = 1,...,k. Here S is the set \(\bigcup\limits_{i = 1}^n {V_{\pi (i)} ,G_{S_n } } \)is the subgraph of G induced by the subset V(G) — Sn of V(G) and γ(G)Sn is the group of permutations in γ(G) which fix each vertex in Sn, considered as acting on V(G) − Sn. The stability index of G, s.i. (G), is the maximum cardinality of a partial stabilising sequence for G; thus s.i. (G) = 0 if and only if G is not semi-stable (see [6]) and s.i. (G) = ¦v(G)¦ if and only if G is stable (see [4]).