Destroying automorphisms by fixing nodes

• Published in: Discrete mathematics Vol. 306; no. 24; pp. 3244 - 3252
• Main Authors: Erwin, David, Harary, Frank
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 28-12-2006; Elsevier
• Subjects: Automorphism group; Combinatorics; Combinatorics. Ordered structures; Exact sciences and technology; Fixing number; Graph theory; Mathematics; Sciences and techniques of general use; Symmetry breaking; Fixing number; Automorphism group; Symmetry breaking; Observation; Graph theory; Necessary and sufficient condition
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/j.disc.2006.06.004

The fixing number of a graph
G is the minimum cardinality of a set
S
⊂
V
(
G
)
such that every nonidentity automorphism of
G moves at least one member of
S, i.e., the automorphism group of the graph obtained from
G by fixing every node in
S is trivial. We provide a formula for the fixing number of a disconnected graph in terms of the fixing numbers of its components and make some observations about graphs with small fixing numbers. We determine the fixing number of a tree and find a necessary and sufficient condition for a tree to have fixing number 1.