Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-23T16:24:09.662Z Has data issue: false hasContentIssue false
  • English
  • Français

Semi-stable and stable cacti

Published online by Cambridge University Press:  09 April 2009

K. L. McAvaney
K. L. McAvaney
Affiliation:
Mathematics Department, Gordon Institute of TechnologyGeelong, Vic. 3220, Australia.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 20 , Issue 4 , November 1975 , pp. 419 - 430
Copyright
Copyright © Australian Mathematical Society 1975

References

Harary, F. (1969), Graph Theory (Addison-Wesley, 1969).CrossRefGoogle Scholar
Heffernan, P. (1972), Trees (Masters Dissertation, University of Canterbury, New Zealand).Google Scholar
Holton, D. A. (1973), ‘Two applications of semi-stable graphs’, Discrete Math. 4, 151–158.CrossRefGoogle Scholar
Holton, D. A. (1973 a), ‘A report on stable graphs’, J. Austral. Math. Soc. 15, 163–171.CrossRefGoogle Scholar
Holton, D. A. (1973 b), ‘Stable trees’, J. Austral. Math. Soc. 15, 476–481.CrossRefGoogle Scholar
Holton, D. A. and Grant, D. D. (1975), ‘Regular graphs and stability’, J. Austral. Math. Soc. 20, 377–384.CrossRefGoogle Scholar
Holton, D. A. and Grant, D. D., (to appear), ‘Products of graphs and stability’, J. Combinatorial Theory.Google Scholar
McAvaney, K. L., Grant, D. D. and Holton, D. A. (1974), ‘Stable and semi-stable unicyclic graphs’, Discrete Math. 9, 277–288.CrossRefGoogle Scholar
Robertson, N. and Zimmer, J. A. (1972), ‘Automorphism of subgraphs obtained by deleting a pendant vertex’, J. Combinatorial Theory Ser. B. 12, 169–173.CrossRefGoogle Scholar