Abstract
A subset U of vertices of a graph G is called a determining set if every automorphism of G is uniquely determined by its action on the vertices of U. A subset W is called a resolving set if every vertex in G is uniquely determined by its distances to the vertices of W. Determining (resolving) sets are said to have the exchange property in G if whenever S and R are minimal determining (resolving) sets for G and \({r\in R}\) , then there exists \({s\in S}\) so that \({S-\{s\} \cup \{r\}}\) is a minimal determining (resolving) set. This work examines graph families in which these sets do, or do not, have the exchange property. This paper shows that neither determining sets nor resolving sets have the exchange property in all graphs, but that both have the exchange property in trees. It also gives an infinite graph family (n-wheels where n ≥ 8) in which determining sets have the exchange property but resolving sets do not. Further, this paper provides necessary and sufficient conditions for determining sets to have the exchange property in an outerplanar graph.
Similar content being viewed by others
References
Albertson, M.O., Boutin, D.L.: Using determining sets to distinguish Kneser graphs. Electron. J. Combin. 14(1):Research Paper 20 (electronic) (2007)
Albertson M.O., Boutin D.L.: Automorphisms and distinguishing numbers of geometric cliques. Discrete Comput. Geom. 39(1), 778–785 (2008)
Babai L.: On the complexity of canonical labeling of strongly regular graphs. SIAM J. Comput. 9(1), 212–216 (1980)
Boutin D.L.: The determining number of a Cartesian product. J. Graph Theory 61, 77–87 (2009)
Boutin, D.L.: Identifying graph automorphisms using determining sets. Electron. J. Combin. 13(1):Research Paper 78 (electronic) (2006)
Buczkowski P.S., Chartrand G., Poisson C., Zhang P.: On k-Dimensional graphs and their bases. Period. Math. Hungar. 46(1), 9–15 (2003)
Cáceres J., Hernando C., Mora M., Pelayo I.M., Puertas M.L., Seara C., Wood D.R.: On the metric dimension of Cartesian products of graphs. SIAM J. Discrete Math. 21, 423–441 (2007)
Chartrand G., Eroh L., Johnson M.A., Oellermann O.R.: Resolvability in graphs and the metric dimension of a graph. Discrete Appl. Math. 105(1–3), 99–113 (2000)
Colbourn C.J., Booth K.S.: Linear time automorphism algorithms for trees, interval graphs, and planar graphs. SIAM J. Comput. 10(1), 203–225 (1981)
Colbourn C.J., Slater P.J., Stewart L.K.: Locating dominating sets in series parallel networks. Congr. Numer. 56, 135–162 (1987). Sixteenth Manitoba conference on numerical mathematics and computing (Winnipeg, Man., 1986)
Dixon J.D., Mortimer B.: Permutation Groups Graduate. Texts in Mathematics, vol. 163. Springer, New York (1996)
Erwin D.J., Harary F.: Destroying automorphisms by fixing nodes. Discrete Math. 306(24), 3244–3252 (2006)
Rall, D.F., Slater, P.J.: On location-domination numbers for certain classes of graphs. In: Proceedings of the Fifteenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Baton Rouge, La., 1984), vol. 45, pp. 97–106 (1984)
Slater, P.J.: Leaves of trees. In: Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), pp. 549–559. Congressus Numerantium, No. XIV, Winnipeg, Man., 1975. Utilitas Math
West D.B.: Introduction to Graph Theory. Prentice Hall Inc., Upper Saddle River (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Boutin, D.L. Determining Sets, Resolving Sets, and the Exchange Property. Graphs and Combinatorics 25, 789–806 (2009). https://doi.org/10.1007/s00373-010-0880-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-010-0880-6