Fixing Number in Graphs and Groups

• Fixing number is the minimum size of a vertex subset that, when fixed, eliminates all nontrivial automorphisms in a graph.
• The concept extends to groups where the fixing set comprises all fixing numbers from graphs whose automorphism groups equal a given finite group.
• Analyses show that graph constructions and algorithmic approaches exploit fixing number properties to balance symmetry-breaking with computational complexity.

Searching arXiv for recent and foundational papers on fixing number in graphs and groups. In graph theory, the fixing number is the minimum cardinality of a subset of vertices whose pointwise stabilizer in the automorphism group is trivial. If $G$ is a finite graph and $S\subseteq V(G)$, then $S$ is a fixing set when the only automorphism fixing every vertex of $S$ is the identity, and $\fix(G)$ is the minimum size of such a set. At the level of finite groups, the associated fixing set of a group $\Gamma$ is the set of all fixing numbers realized by finite graphs with automorphism group $\Gamma$. The parameter is therefore a symmetry-breaking invariant, and, from the permutation-group viewpoint, it is a base-size parameter for a faithful action (Gibbons et al., 2018, Arvind et al., 2016).

1. Definition and basic examples

Let $G=(V,E)$ be a finite graph with automorphism group $\Aut(G)$. For $S\subseteq V$, the pointwise stabilizer is

$S\subseteq V(G)$0

A fixing set is a subset $S\subseteq V(G)$1 with $S\subseteq V(G)$2, and the fixing number is

$S\subseteq V(G)$3

In colored-graph language, the same definition is used for a colored graph $S\subseteq V(G)$4, with $S\subseteq V(G)$5 restricted to color-preserving automorphisms; the uncolored case is the special case in which all vertices have the same initial color (Arvind et al., 2016).

The parameter measures how many vertices must be “pinned down” to eliminate all nontrivial symmetries. It satisfies $S\subseteq V(G)$6, and $S\subseteq V(G)$7 exactly for rigid graphs (Fazil et al., 2016).

Graph $S\subseteq V(G)$8	Value of $S\subseteq V(G)$9	Source
$S$0	$S$1	(Javaid et al., 2015)
$S$2 for $S$3	$S$4	(Javaid et al., 2015)
$S$5 for $S$6	$S$7	(Javaid et al., 2015)

These examples already exhibit the spectrum from maximal symmetry to near-rigidity. Complete graphs require fixing all but one vertex, whereas a path requires only one endpoint, and a cycle requires two nonadjacent vertices to break both rotational and reflectional symmetry (Javaid et al., 2015).

2. Action-theoretic interpretation and general bounds

The fixing number is equivalent to the determining-set parameter: a subset is a fixing set if and only if any two automorphisms agreeing on that subset agree on all of $S$8 (Gibbons et al., 2018). This identifies the notion with a standard base-size concept from permutation-group theory. In the matroid setting, the same point of view is explicit: if $S$9 is a matroid with automorphism group $S$0 acting faithfully on its ground set $S$1, then $S$2 is exactly the base size of that action (Gordon et al., 2013).

A basic comparison is with the distinguishing number $S$3. If $S$4 is a fixing set of size $S$5, then labeling the vertices in $S$6 bijectively by $S$7 and all remaining vertices by one new label yields a distinguishing labeling with $S$8 labels, so

$S$9

Conversely, any distinguishing labeling yields a fixing set of size at most $\fix(G)$0, so the two parameters coincide up to an additive constant (Gibbons et al., 2018).

Group-theoretic bounds are also available. If $\fix(G)$1 and $\fix(G)$2 denotes the subgroup-chain length of $\fix(G)$3, then

$\fix(G)$4

Moreover, if $\fix(G)$5, then $\fix(G)$6, so $\fix(G)$7 is bounded above by the number of prime factors of $\fix(G)$8 counted with multiplicity (Gibbons et al., 2018). In the matroid setting, if $\fix(G)$9, $\Gamma$0, and $\Gamma$1 is the largest orbit size under $\Gamma$2, then

$\Gamma$3

together with the orbit lower bound $\Gamma$4 (Gordon et al., 2013).

These bounds situate fixing number between combinatorial orbit structure and group size. They are particularly useful when exact computation is difficult, which is typical outside highly structured graph classes.

3. Fixing sets of groups and Hamiltonian-group realizations

For a finite group $\Gamma$5, the fixing set is

$\Gamma$6

Every nontrivial finite group occurs as the automorphism group of a Cayley graph or Frucht graph, and in such realizations a single vertex suffices, so $\Gamma$7 for every nontrivial finite $\Gamma$8 (Gibbons et al., 2018).

For finite abelian groups, the structure is completely determined. If $\Gamma$9 is abelian with $\Gamma$0 elementary divisors, then

$\Gamma$1

This gives a full initial segment of the positive integers, with endpoint equal to the number of elementary divisors (Gibbons et al., 2018).

A more recent classification concerns finite Hamiltonian groups. A finite non-abelian group $\Gamma$2 is Hamiltonian if all of its subgroups are normal, and every finite Hamiltonian group has the form

$\Gamma$3

where $\Gamma$4 is the quaternion group of order $\Gamma$5, $\Gamma$6 is an abelian group of odd exponent, and $\Gamma$7 is an elementary abelian $\Gamma$8-group (Sahu et al., 29 Jan 2026). For minimal graph realizations, if $\Gamma$9, then the least number of vertices of a graph $G=(V,E)$0 with $G=(V,E)$1 is

$G=(V,E)$2

and more generally, if $G=(V,E)$3 where $G=(V,E)$4 is periodic abelian with no element of order $G=(V,E)$5, then

$G=(V,E)$6

The fixing-set classification is equally explicit: if $G=(V,E)$7 is a finite Hamiltonian group and $G=(V,E)$8 has $G=(V,E)$9 elementary divisors, then

$\Aut(G)$0

In particular,

$\Aut(G)$1

and for $\Aut(G)$2 one obtains $\Aut(G)$3 (Sahu et al., 29 Jan 2026).

These results show that fixing sets of groups are not arbitrary subsets of $\Aut(G)$4. For abelian and Hamiltonian groups they are full initial segments, with endpoints controlled by the elementary-divisor structure of the abelian component.

4. Behavior under graph constructions

Fixing number behaves predictably under several graph products, although the exact behavior depends strongly on the construction.

For the composition product $\Aut(G)$5, if $\Aut(G)$6 is connected of order $\Aut(G)$7 and $\Aut(G)$8 is arbitrary of order $\Aut(G)$9 with components $S\subseteq V$0, then

$S\subseteq V$1

If $S\subseteq V$2 is connected, then the lower bound is exact: $S\subseteq V$3 For the corona product, if $S\subseteq V$4 is connected of order $S\subseteq V$5 and $S\subseteq V$6 is arbitrary, then

$S\subseteq V$7

and in the non-asymmetric case one has the simpler formula $S\subseteq V$8 (Javaid et al., 2015).

For functigraphs $S\subseteq V$9, formed from two copies $S\subseteq V(G)$00 of a graph $S\subseteq V(G)$01 and a function $S\subseteq V(G)$02, the global range is sharp: $S\subseteq V(G)$03 for connected $S\subseteq V(G)$04 of order $S\subseteq V(G)$05. The lower bound occurs for $S\subseteq V(G)$06 with a suitable $S\subseteq V(G)$07, and the upper bound occurs for $S\subseteq V(G)$08 with constant $S\subseteq V(G)$09. The constructions also show that $S\subseteq V(G)$10 and $S\subseteq V(G)$11 can both be made arbitrarily large (Fazil et al., 2016).

For the co-normal product $S\subseteq V(G)$12 of graphs of orders $S\subseteq V(G)$13, the sharp general bounds are

$S\subseteq V(G)$14

The upper bound is attained when both factors are complete or both are null, while the lower bound is attained in several no-twin/no-dominator regimes (Rehman et al., 2017).

A recurring theme across these constructions is that local symmetry classes—twins, false twins, fibers of a function, or repeated layers—force additive or multiplicative contributions to the fixing number. This suggests that fixing number is particularly sensitive to replicated local structure.

5. Algorithmic and parameterized complexity

The decision version of the problem for colored graphs is often written as $S\subseteq V(G)$15-RIGID: given a colored graph $S\subseteq V(G)$16 and integer $S\subseteq V(G)$17, decide whether there exists $S\subseteq V(G)$18 with $S\subseteq V(G)$19 such that $S\subseteq V(G)$20 is trivial, equivalently whether $S\subseteq V(G)$21 (Arvind et al., 2016). This “forward” parameterization is hard: $S\subseteq V(G)$22-RIGID is MINI[1]-hard, even when $S\subseteq V(G)$23 is an elementary abelian $S\subseteq V(G)$24-group (Arvind et al., 2016).

The dual parameterization asks whether there exists a fixing set of size at least $S\subseteq V(G)$25, where $S\subseteq V(G)$26. The corresponding group-theoretic analogue, $S\subseteq V(G)$27-BASE-SIZE, admits an algorithm running in time

$S\subseteq V(G)$28

together with a kernel of size $S\subseteq V(G)$29. For graphs, the dual problem $S\subseteq V(G)$30-RIGID can be solved in time

$S\subseteq V(G)$31

The method uses orbit and block decomposition, together with either computation of $S\subseteq V(G)$32 by a graph-isomorphism subroutine or enumeration of automorphisms of support at most $S\subseteq V(G)$33 (Arvind et al., 2016).

The resulting complexity dichotomy is sharp at the level stated in the data: when one seeks a very small fixing set, the problem is “essentially intractable,” whereas fixing almost all vertices becomes feasible for small $S\subseteq V(G)$34 (Arvind et al., 2016). This is a rare instance where two natural parameterizations of the same symmetry-breaking problem have radically different complexity.

6. Variants, extensions, and related notions

A linear-programming relaxation yields the fractional fixing number $S\subseteq V(G)$35. Writing $S\subseteq V(G)$36 for the set of ordered pairs $S\subseteq V(G)$37 of distinct vertices lying in the same nontrivial orbit, and $S\subseteq V(G)$38 for the fixing neighborhood of $S\subseteq V(G)$39, one relaxes the integer program for $S\subseteq V(G)$40 to obtain

$S\subseteq V(G)$41

subject to $S\subseteq V(G)$42 and $S\subseteq V(G)$43 for every $S\subseteq V(G)$44. Always,

$S\subseteq V(G)$45

Moreover, for a nontrivial graph $S\subseteq V(G)$46 on $S\subseteq V(G)$47 vertices, the following are equivalent: $S\subseteq V(G)$48; every vertex has at least one twin; and $S\subseteq V(G)$49 is isomorphic to a generalized lexicographic product $S\subseteq V(G)$50 in which each inner graph is a nontrivial complete graph or a nontrivial empty graph (Benish et al., 2016).

For trees, the fixing number is closely tied to distinguishing colorings. If $S\subseteq V(G)$51 is the distinguishing number and $S\subseteq V(G)$52 the fixing number, then

$S\subseteq V(G)$53

For trees these bounds can be sharpened substantially. Every $S\subseteq V(G)$54-distinguishable tree $S\subseteq V(G)$55 of order $S\subseteq V(G)$56 satisfies

$S\subseteq V(G)$57

and every $S\subseteq V(G)$58-distinguishable tree with $S\subseteq V(G)$59 satisfies

$S\subseteq V(G)$60

with both bounds sharp (Buchanan et al., 25 Mar 2026).

The notion also extends beyond graphs. For matroids, a fixing set is a subset of the ground set whose pointwise stabilizer in $S\subseteq V(G)$61 is trivial, and $S\subseteq V(G)$62 is again the base size of the faithful automorphism action. If $S\subseteq V(G)$63 is a $S\subseteq V(G)$64-connected graph with at least $S\subseteq V(G)$65 vertices, then the cycle matroid $S\subseteq V(G)$66 and the bicircular matroid $S\subseteq V(G)$67 satisfy

$S\subseteq V(G)$68

This follows from the identification of $S\subseteq V(G)$69 with both $S\subseteq V(G)$70 and $S\subseteq V(G)$71 under the stated connectivity hypotheses (Gordon et al., 2013).

The term “fixing number” is also used in distinct, non-equivalent senses in nearby literatures. In graph recoloring, $S\subseteq V(G)$72 denotes the maximum distance from an $S\subseteq V(G)$73-coloring to a proper $S\subseteq V(G)$74-coloring, rather than an automorphism-breaking parameter (Garnero et al., 2016). This suggests that the graph-automorphism fixing number should be interpreted contextually: in combinatorics and permutation-group theory it is a base-size invariant, while in other algorithmic settings the same phrase may refer to a repair or correction distance.