Identifying Graph Automorphisms Using Determining Sets

Abstract

A set of vertices $S$ is a determining set for a graph $G$ if every automorphism of $G$ is uniquely determined by its action on $S$. The determining number of a graph is the size of a smallest determining set. This paper describes ways of finding and verifying determining sets, gives natural lower bounds on the determining number, and shows how to use orbits to investigate determining sets. Further, determining sets of Kneser graphs are extensively studied, sharp bounds for their determining numbers are provided, and all Kneser graphs with determining number $2$, $3,$ or $4$ are given.

Base size, metric dimension and other invariants of groups and graphs

“…The equivalent problem of finding the determining numbers of Kneser graphs has been considered more recently, starting in the 2006 paper of Boutin [19], and continued in the recent work of Cáceres et al [27]. Their methods and results are very similar to Maund's, although were obtained independently, and are expressed in the language of graphs and hypergraphs.…”

“…More specifically, can the gap between the two parameters be made arbitrarily large? This question is asked by Boutin [19] and (implicitly) by Vince [93], while the paper by Cáceres et al [28] is devoted to investigating it. In the same vein, we can ask: for which graphs are the two parameters equal?…”

“…Erwin and Harary [48] used the term fixing set to mean a base for Aut(Γ), and call the base size the fixing number of Γ. In a similar manner, Boutin [19] used the term determining set to mean a base for Aut(Γ), and calls the base size of Aut(Γ) the determining number of Γ. Also, Fijavž and Mohar [55] call the base size of Aut(Γ) the rigidity index of Γ.…”

Fixing Numbers of Graphs and Groups

“…Equivalently, S is a fixing set of the graph G if whenever g ∈ Aut(G) fixes every vertex in S, g is the identity automorphism. A set of vertices S is a determining set of G if whenever two automorphisms g, h ∈ Aut(G) agree on S, then they agree on G, i.e., they are the same automorphism [3]. The following lemma shows that these two definitions are equivalent.…”

On the Distinguishing Number of Functigraphs

“…They have also given a comparison between the distinguishing number and the distinguishing index for a connected graph G of order n ≥ 3. Boutin [7] introduced the concept of determining sets. In [4], Albertson and Boutin proved that a graph is t-distinguishable if and only if it has a determining set that is (t − 1)distinguishable.…”