Restriction Graphs: A Constraint Framework

• Restriction graphs are graph-based formalisms where vertices and edges encode admissibility constraints on combinatorial and statistical objects.
• They underlie state-space generation in restricted Tower of Hanoi, constrained permutation theory, and missing-data models by imposing local move or inequality conditions.
• Applications extend to graph-restrictive permutation groups, graph rewriting systems, and algebraic CSPs, unifying local restrictions with global structural and tractability results.

Searching arXiv for recent and relevant papers on “restriction graphs” and closely related usages. Searching arXiv for: restriction graph permutation graph restrictive graphs restriction species pattern graphs restricted Hanoi graphs. Restriction graphs are graph-based formalisms in which vertices and edges encode admissibility constraints on another mathematical, combinatorial, or statistical object. In different literatures, the constrained object may be a Tower of Hanoi move system, a family of permutations, a missing-data response-pattern model, a class of structures closed under restriction to subobjects, or a graph transformation equipped with application conditions. The common mechanism is that the graph records which local transitions, comparisons, extensions, or subobject selections are permitted, and the global object is then defined or analyzed relative to those permitted local relations (Mehiri, 2023, Tymoshenko et al., 8 Jul 2025, Chen, 2020, Gálvez-Carrillo et al., 2017).

1. Terminological range and common schema

The expression “restriction graph” is used in several non-equivalent but structurally related senses. In the restricted Tower of Hanoi literature, the restriction graph is a digraph $D=(\mathcal M,A(D))$ on the peg set $\mathcal M=\{1,\ldots,m\}$, where an arc $(p,q)\in A(D)$ means that a disc is allowed to move from peg $p$ to peg $q$. In constrained permutation theory, a restriction graph is an oriented graph $G=([n],E)$ on positions, and a permutation $\sigma\in S_n$ satisfies $G$ when every directed edge $u\to v\in E$ imposes the inequality $\sigma_u>\sigma_v$. In missing-data theory, a pattern graph is a directed graph on response patterns, not on variables, and its edges encode how one pattern can “generate” another. In the theory of restriction species, restriction is encoded categorically rather than by a single distinguished graph: a restriction species is a presheaf $\mathcal M=\{1,\ldots,m\}$0 on finite sets and injections, so that structures can be restricted to subsets. In a further combinatorial usage, horizontal visibility graphs built from restricted growth sequences are “restriction-based” because the underlying sequences satisfy the restricted growth condition rather than ranging over arbitrary words (Mehiri, 2023, Tymoshenko et al., 8 Jul 2025, Chen, 2020, Gálvez-Carrillo et al., 2017, Mansour et al., 2019).

Despite this variation, the same formal pattern recurs. A base graph, digraph, or categorical graph-like device specifies a local admissibility relation. Valid global objects are then those respecting that relation: legal Hanoi states, linear extensions compatible with directed inequalities, identifiable missing-data models, induced substructures, or graph-rewriting derivations compatible with application conditions. This suggests that “restriction graph” is best understood as a family of graph-encoded constraint mechanisms rather than as a single canonical definition.

2. State-space and order-theoretic restriction graphs

In restricted Hanoi theory, the movement digraph $\mathcal M=\{1,\ldots,m\}$1 determines the entire state graph $\mathcal M=\{1,\ldots,m\}$2. A state is an $\mathcal M=\{1,\ldots,m\}$3-tuple $\mathcal M=\{1,\ldots,m\}$4, where $\mathcal M=\{1,\ldots,m\}$5 is the peg holding disc $\mathcal M=\{1,\ldots,m\}$6, so $\mathcal M=\{1,\ldots,m\}$7 has size $\mathcal M=\{1,\ldots,m\}$8. Two states $\mathcal M=\{1,\ldots,m\}$9 are adjacent only if there exists a unique $(p,q)\in A(D)$0 such that only disc $(p,q)\in A(D)$1 moves, disc $(p,q)\in A(D)$2 is topmost on its source peg, disc $(p,q)\in A(D)$3 can be placed on the destination peg, and $(p,q)\in A(D)$4. The paper turns this into an explicit generation test by computing $(p,q)\in A(D)$5, rejecting unless $(p,q)\in A(D)$6, checking $(p,q)\in A(D)$7, and then verifying the smaller-disc constraints. The straightforward generation algorithm runs in time $(p,q)\in A(D)$8. The same framework specializes to the classical unrestricted, linear, cyclic, and star variants by replacing condition $(p,q)\in A(D)$9 with a simple arithmetic test. The paper also proves

$p$0

for the number of arcs arising from moves of disc $p$1, and

$p$2

for the total number of directed edges. Thus the restriction graph enters both the local move rule and the global edge count, with the latter depending on $p$3 only through $p$4 (Mehiri, 2023).

In permutation theory, the restriction graph $p$5 defines the family $p$6 of all permutations satisfying the edge inequalities $p$7. Reachability induces a partial order

$p$8

and any $p$9-admissible restriction graph must be acyclic, since an oriented cycle would force a contradiction $q$0. The metric geometry of $q$1 is then controlled by this induced poset. For the $q$2-metric,

$q$3

the paper proves the exact diameter formula

$q$4

and gives a greedy source-removal algorithm constructing optimal permutations. For the Kendall–Tau metric,

$q$5

the combinatorial upper bound is the number of incomparable pairs in the induced poset $q$6, and this bound is achieved if and only if $q$7. In that case, extremal permutations form a minimal realizer of the poset. The same framework subsumes descent-set families and Hessenberg inversion families, yielding explicit formulas and efficient algorithms for metric diameters (Tymoshenko et al., 8 Jul 2025).

These two settings illustrate complementary forms of restriction-graph behavior. In restricted Hanoi, the graph constrains legal transitions in a state space. In permutation families, it constrains admissible total orders. In both cases, a local digraph rule propagates to exact global structural and enumerative consequences.

3. Local symmetry and type-restriction graphs

A different restriction mechanism appears in the theory of graph-restrictive permutation groups. Let $q$8 be a finite connected $q$9-vertex-transitive graph and $G=([n],E)$0. If the permutation group induced by $G=([n],E)$1 on the neighborhood $G=([n],E)$2,

$G=([n],E)$3

is permutation isomorphic to a fixed group $G=([n],E)$4, then $G=([n],E)$5 is locally-$G=([n],E)$6. The permutation group $G=([n],E)$7 is graph-restrictive if there exists a constant $G=([n],E)$8 such that for every locally-$G=([n],E)$9 pair $\sigma\in S_n$0 and every vertex $\sigma\in S_n$1,

$\sigma\in S_n$2

The paper proves that an intransitive permutation group is graph-restrictive if and only if it is semiregular. The easy direction shows that if $\sigma\in S_n$3 is semiregular of degree $\sigma\in S_n$4, then every locally-$\sigma\in S_n$5 pair satisfies $\sigma\in S_n$6. The converse is constructive: if $\sigma\in S_n$7 is intransitive and not semiregular, the authors build finite locally-$\sigma\in S_n$8 graphs with vertex-stabilizers of unbounded size, ultimately obtaining examples with

$\sigma\in S_n$9

Although this usage concerns graph-restrictive groups rather than a graph explicitly named a restriction graph, it exhibits the same principle: fixing a local graph action imposes, or fails to impose, global bounds on stabilizers (Spiga et al., 2012).

In the algebraic CSP literature, the associated graph $G$0 of a finite algebra $G$1 or relational structure $G$2 is explicitly a graph of restricted local types. Its vertices are the elements of the universe, and an edge $G$3 is present when the subalgebra generated by $G$4 admits a “good” term operation on some quotient: semilattice, majority, or affine. An algebra is $G$5-restricted, for $G$6, if every edge of $G$7 has a type in $G$8. The paper distinguishes thick edges, defined via quotients $G$9, from thin edges, which refine local connectivity arguments. One of its main stability results states that if a finite idempotent algebra has only edge types from a set $u\to v\in E$0, then every finite algebra in the variety generated by it is also $u\to v\in E$1-restricted. This restricted-type graph controls complexity consequences: if every edge is majority or affine, then the algebra has few subpowers and $u\to v\in E$2 is polynomial-time solvable; if affine edges are absent, bounded width holds; and if only semilattice and majority edges occur, then $u\to v\in E$3 has bounded width, in fact width $u\to v\in E$4 (Bulatov, 2020).

Taken together, these theories treat graphs as repositories of local structural restrictions. In one case the restriction is on local permutation action; in the other it is on local term-operation type. In both, the graph serves as a compressed certificate from which global regularity or algorithmic tractability is derived.

4. Restriction to subgraphs, extension problems, and effective couplings

Restriction graphs also arise when the principal operation is passage to a subgraph. In the GKM framework, a labeled graph $u\to v\in E$5 carries a homogeneous linear polynomial label $u\to v\in E$6 on each directed edge, with antisymmetry $u\to v\in E$7. Its graph cohomology is

$u\to v\in E$8

For a GKM $u\to v\in E$9-variety $\sigma_u>\sigma_v$0, the GKM theorem gives

$\sigma_u>\sigma_v$1

so restriction of equivariant cohomology to an invariant subvariety becomes an extension problem from an induced subgraph $\sigma_u>\sigma_v$2. If $\sigma_u>\sigma_v$3, then $\sigma_u>\sigma_v$4 extends to $\sigma_u>\sigma_v$5 if and only if for any two neighbors $\sigma_u>\sigma_v$6 of $\sigma_u>\sigma_v$7, with edge labels $\sigma_u>\sigma_v$8,

$\sigma_u>\sigma_v$9

This is the graph-theoretic Chinese remainder criterion. A pair $\mathcal M=\{1,\ldots,m\}$00 is a Chinese remainder pair when every class on $\mathcal M=\{1,\ldots,m\}$01 extends, and successive one-vertex extensions yield surjectivity of $\mathcal M=\{1,\ldots,m\}$02. The paper gives geometric sufficient conditions for Euclidean embedded graphs, then applies them to smooth projective toric varieties through $\mathcal M=\{1,\ldots,m\}$03-face connected induced subgraphs, and to Bruhat graphs of flag varieties (Carrell et al., 2020).

A probabilistic analogue appears in reinforced processes restricted to a subset of vertices. For a finite graph $\mathcal M=\{1,\ldots,m\}$04 with partition $\mathcal M=\{1,\ldots,m\}$05, the restriction of a jump process $\mathcal M=\{1,\ldots,m\}$06 to $\mathcal M=\{1,\ldots,m\}$07 is

$\mathcal M=\{1,\ldots,m\}$08

After removal of self-loops one obtains $\mathcal M=\{1,\ldots,m\}$09. For the vertex-reinforced jump process in exchangeable time scale, the effective weights on $\mathcal M=\{1,\ldots,m\}$10 are

$\mathcal M=\{1,\ldots,m\}$11

with self-loop-free version

$\mathcal M=\{1,\ldots,m\}$12

The main theorem states that $\mathcal M=\{1,\ldots,m\}$13 and $\mathcal M=\{1,\ldots,m\}$14 are mixtures of VRJPs on $\mathcal M=\{1,\ldots,m\}$15 with these random effective weights. The paper then applies this restriction mechanism to subdivided graphs, where every edge is replaced by a chain of $\mathcal M=\{1,\ldots,m\}$16 edges, proving a recursion

$\mathcal M=\{1,\ldots,m\}$17

for effective weights and deducing positive recurrence results for suitable restricted discrete-time processes, as well as analogous results for edge-reinforced random walk (Disertori et al., 2024).

These two lines of work make restriction to a subgraph an operative transformation rather than a passive deletion. In graph cohomology it becomes an extension criterion; in reinforced processes it becomes an exact renormalization of transition structure through effective couplings.

5. Pattern graphs, application conditions, and movable restrictions

In nonmonotone missing-data theory, restriction graphs appear as pattern graphs. If $\mathcal M=\{1,\ldots,m\}$18 is the full variable vector and $\mathcal M=\{1,\ldots,m\}$19 is the response-pattern indicator, then the nodes of the graph are the possible response patterns $\mathcal M=\{1,\ldots,m\}$20. For patterns $\mathcal M=\{1,\ldots,m\}$21, the relation

$\mathcal M=\{1,\ldots,m\}$22

means that $\mathcal M=\{1,\ldots,m\}$23 observes everything observed by $\mathcal M=\{1,\ldots,m\}$24, and possibly more. A regular pattern graph satisfies: $\mathcal M=\{1,\ldots,m\}$25 is the only source node, and if $\mathcal M=\{1,\ldots,m\}$26, then $\mathcal M=\{1,\ldots,m\}$27. Under this structure, the selection-odds restriction

$\mathcal M=\{1,\ldots,m\}$28

defines a nonparametrically identifiable/saturated model, often of missing-not-at-random type. Writing

$\mathcal M=\{1,\ldots,m\}$29

the complete-case probability

$\mathcal M=\{1,\ldots,m\}$30

is identified recursively through

$\mathcal M=\{1,\ldots,m\}$31

The same restriction graph admits an equivalent pattern-mixture formulation,

$\mathcal M=\{1,\ldots,m\}$32

and supports IPW, regression-adjustment or imputation-based, and multiply robust semiparametric estimators, together with efficient influence function calculations (Chen, 2020).

In Matrix Graph Grammars, restrictions are graph constraints and application conditions attached to productions. A graph constraint is

$\mathcal M=\{1,\ldots,m\}$33

where $\mathcal M=\{1,\ldots,m\}$34 is a diagram of graphs and partial injective morphisms, and $\mathcal M=\{1,\ldots,m\}$35 is a logical formula using predicates

$\mathcal M=\{1,\ldots,m\}$36

Application conditions are specialized graph constraints for a production $\mathcal M=\{1,\ldots,m\}$37 with nihilation matrix $\mathcal M=\{1,\ldots,m\}$38. The paper’s central result is that preconditions and postconditions are transformable into one another: any consistent precondition is equivalent to some consistent postcondition and vice versa. The corresponding condition can also be moved along a production sequence, a phenomenon called restriction delocalization. This is expressed by sequence equations such as

$\mathcal M=\{1,\ldots,m\}$39

when the relevant checking productions and the rule are sequentially independent. The same machinery is then used to encode multidigraph rewriting inside simple digraph rewriting by adding graph constraints that enforce the intended edge-as-multinode representation (0912.2160).

Both pattern graphs and MGG application conditions treat restrictions as first-class formal objects. In the former, the restriction graph identifies the full-data law. In the latter, restrictions are movable algebraic components of rewriting, not merely external side conditions.

6. Containment restrictions, hereditary classes, and categorical generalization

Containment-restricted interval graphs provide a classical graph-theoretic example of restriction by representation. A graph is $\mathcal M=\{1,\ldots,m\}$40-improper if it has an interval representation in which no interval contains more than $\mathcal M=\{1,\ldots,m\}$41 other intervals; the proper interval graphs are exactly $\mathcal M=\{1,\ldots,m\}$42. This notion is dual to the earlier $\mathcal M=\{1,\ldots,m\}$43-proper condition, where no interval is properly contained in more than $\mathcal M=\{1,\ldots,m\}$44 other intervals. For a representation $\mathcal M=\{1,\ldots,m\}$45, the impropriety of a vertex interval $\mathcal M=\{1,\ldots,m\}$46 is the number of intervals contained in it, the impropriety of the representation is the maximum of these values, and the impropriety of the graph is the minimum over all representations. The paper defines the graph-theoretic weight $\mathcal M=\{1,\ldots,m\}$47 of a vertex $\mathcal M=\{1,\ldots,m\}$48 as the sum of the orders of the $\mathcal M=\{1,\ldots,m\}$49 smallest non-exterior local components of $\mathcal M=\{1,\ldots,m\}$50, proves

$\mathcal M=\{1,\ldots,m\}$51

and calls a graph balanced when equality holds. In a connected interval graph, the vertices of positive weight induce a disjoint union of paths. For the balanced $\mathcal M=\{1,\ldots,m\}$52-critical case, the paper gives a complete structural classification: $\mathcal M=\{1,\ldots,m\}$53 is $\mathcal M=\{1,\ldots,m\}$54-critical and balanced if and only if it is isomorphic to one of the $\mathcal M=\{1,\ldots,m\}$55 constructions with the stated clique-component conditions (Beyerl et al., 2011).

Hereditary graph theory yields a limit-theoretic version of restriction by forbidden pattern. A graph class $\mathcal M=\{1,\ldots,m\}$56 is hereditary if it is closed under induced subgraphs, equivalently

$\mathcal M=\{1,\ldots,m\}$57

For hereditary $\mathcal M=\{1,\ldots,m\}$58 and a graph limit $\mathcal M=\{1,\ldots,m\}$59, the paper proves

$\mathcal M=\{1,\ldots,m\}$60

and also

$\mathcal M=\{1,\ldots,m\}$61

If $\mathcal M=\{1,\ldots,m\}$62 denotes the class of graphs with no induced subgraph from $\mathcal M=\{1,\ldots,m\}$63, then the corresponding graph-limit class is

$\mathcal M=\{1,\ldots,m\}$64

This framework recovers triangle-free, chordal, cograph, unit interval, threshold, and line graph limits as restriction classes defined by vanishing subgraph densities, and in some cases yields explicit limit structure, such as the split-graphon form with a zero block and a one block, or the identity $\mathcal M=\{1,\ldots,m\}$65 for line graph limits and disjoint clique graph limits (Janson, 2011).

At a categorical level, restriction species abstract the operation of passing to subobjects. An ordinary restriction species is a presheaf

$\mathcal M=\{1,\ldots,m\}$66

so that for an injection $\mathcal M=\{1,\ldots,m\}$67, an $\mathcal M=\{1,\ldots,m\}$68-structure on $\mathcal M=\{1,\ldots,m\}$69 restricts to one on $\mathcal M=\{1,\ldots,m\}$70. Finite graphs are a principal example: $\mathcal M=\{1,\ldots,m\}$71 is the groupoid of graph structures on vertex set $\mathcal M=\{1,\ldots,m\}$72, and restriction is induced subgraph. Such species naturally induce decomposition spaces, and their incidence coalgebras recover Schmitt-type formulas. For graphs,

$\mathcal M=\{1,\ldots,m\}$73

Directed restriction species generalize this to presheaves on finite posets and convex maps, so restriction is to convex subposets rather than arbitrary subsets. The associated coalgebra becomes

$\mathcal M=\{1,\ldots,m\}$74

a non-cocommutative cut formula underlying examples such as rooted forests and directed graphs (Gálvez-Carrillo et al., 2017).

A plausible implication is that the most stable meaning of “restriction graph” is methodological rather than lexical. Whether the setting is interval containment, forbidden induced subgraphs, induced-subobject coalgebras, or order-compatible cuts, the graph functions as a carrier of admissibility data from which one derives structure, classification, or invariants.