VC-Tendency: Thresholds in Graphs and Learning

• VC-Tendency is a phenomenon where VC-theoretic complexity parameters act as thresholds influencing structural, algorithmic, and statistical behaviors in graphs and networks.
• It governs transitions such as sub-linear versus linear twin-width and logarithmic versus polynomial identifying code sizes across hereditary graph classes.
• The concept bridges graph theory, machine learning, and model theory by revealing scaling laws and threshold behaviors that affect optimization and tractability.

Searching arXiv for the cited papers and the phrase "VC-tendency" to ground the article in the relevant literature. Searching for "The Twin-Width of Graphs of Bounded VC-Dimension" and related VC-tendency papers. VC-tendency denotes a recurring phenomenon in which VC-theoretic complexity parameters act as threshold variables for structural, algorithmic, and statistical behavior. In the cited literature, the phenomenon appears in several technically distinct forms: bounded versus unbounded VC-dimension in hereditary graph classes determines whether twin-width is sub-linear or linear and whether identifying codes are polynomially large or logarithmic; bounded color complexity or bounded graph order constrains the VC dimension of graph neural networks through Weisfeiler–Leman distinguishability; bounded primal or dual VC-dimension does not by itself guarantee parameterized tractability for Hitting Set; and, in model theory, VC density tracks dp-rank up to an additive constant and exhibits essentially linear growth in the number of parameters (Biedl et al., 19 Jun 2026, Bousquet et al., 2014, Morris et al., 2023, Johnson, 2011, Bringmann et al., 2015).

1. Formal meanings of VC-based complexity

The term combines several related notions rather than a single invariant. In graph theory, one definition starts from open neighborhoods. For a simple graph $G=(V,E)$, let

$\mathcal{F}(G)=\{N(v):v\in V\}.$

A subset $S\subseteq V$ is shattered by $\mathcal{F}(G)$ if for every $A\subseteq S$ there is some $N(v)\in\mathcal{F}(G)$ with $N(v)\cap S=A$. The VC-dimension of $G$, denoted $vcdim(G)$, is the maximum size of a shattered subset of $V$ (or $\mathcal{F}(G)=\{N(v):v\in V\}.$0 if arbitrarily large shattered sets exist) (Biedl et al., 19 Jun 2026).

For identifying codes, the relevant hypergraph is built from closed neighborhoods. If $\mathcal{F}(G)=\{N(v):v\in V\}.$1 is twin-free, an identifying code is a set $\mathcal{F}(G)=\{N(v):v\in V\}.$2 such that

$\mathcal{F}(G)=\{N(v):v\in V\}.$3

The closed-neighborhood hypergraph is

$\mathcal{F}(G)=\{N(v):v\in V\}.$4

and the VC-dimension of a hereditary class $\mathcal{F}(G)=\{N(v):v\in V\}.$5 is

$\mathcal{F}(G)=\{N(v):v\in V\}.$6

This change from open to closed neighborhoods is technically important, but the same threshold logic reappears (Bousquet et al., 2014).

In learning theory, VC-dimension is defined through growth functions. For a real-valued hypothesis class $\mathcal{F}(G)=\{N(v):v\in V\}.$7 on domain $\mathcal{F}(G)=\{N(v):v\in V\}.$8,

$\mathcal{F}(G)=\{N(v):v\in V\}.$9

The class shatters $S\subseteq V$0 if $S\subseteq V$1. For real-valued classes one often passes to the pseudodimension via

$S\subseteq V$2

and writes $S\subseteq V$3 for this pseudodimension (Sepliarskaia et al., 2024).

In model theory, VC behavior is captured by growth exponents rather than shattering numbers alone. For a partitioned formula $S\subseteq V$4 and finite $S\subseteq V$5,

$S\subseteq V$6

The VC-density of $S\subseteq V$7 is

$S\subseteq V$8

and

$S\subseteq V$9

This is paired with dp-rank through ICT-patterns and the quantity $\mathcal{F}(G)$0 (Johnson, 2011).

A further variant occurs in hypergraph algorithms. For a set system $\mathcal{F}(G)$1, VC-dimension is the largest $\mathcal{F}(G)$2 such that the trace

$\mathcal{F}(G)$3

realizes all subsets of $\mathcal{F}(G)$4. One also studies the dual system and the shatter function

$\mathcal{F}(G)$5

as well as finer $\mathcal{F}(G)$6-system constraints (Bringmann et al., 2015).

2. Hereditary graph classes: the threshold between linear and sub-linear twin-width

In structural graph theory, VC-tendency appears as a threshold for twin-width growth. Twin-width is defined through contraction sequences in trigraphs whose edges are colored black or red. A contraction identifies two vertices $\mathcal{F}(G)$7 into a single new vertex $\mathcal{F}(G)$8; every edge from $\mathcal{F}(G)$9 or $A\subseteq S$0 to a third vertex $A\subseteq S$1 becomes black if both $A\subseteq S$2 and $A\subseteq S$3 were black, and red otherwise, including the case where one was a non-edge. The twin-width $A\subseteq S$4 is the minimum integer $A\subseteq S$5 such that $A\subseteq S$6 admits a contraction sequence in which every intermediate trigraph has maximum red degree at most $A\subseteq S$7 (Biedl et al., 19 Jun 2026).

The decisive result is that bounded VC-dimension forces sub-linear twin-width. If $A\subseteq S$8, then for every $A\subseteq S$9-vertex graph $N(v)\in\mathcal{F}(G)$0 with $N(v)\in\mathcal{F}(G)$1,

$N(v)\in\mathcal{F}(G)$2

and in particular there is some constant $N(v)\in\mathcal{F}(G)$3 such that

$N(v)\in\mathcal{F}(G)$4

This gives a general sub-linear upper bound for every bounded-VC-dimension graph class (Biedl et al., 19 Jun 2026).

The converse threshold is hereditary-class theoretic. A classical result reported there states that a hereditary class has bounded VC-dimension if and only if it excludes, as induced subgraphs, all large split graphs, all large bipartite graphs, and all large co-bipartite graphs. Since each of these three types can have linear twin-width by modifying conference-graph constructions, any hereditary class of unbounded VC-dimension contains graphs of twin-width $N(v)\in\mathcal{F}(G)$5. The resulting equivalence is the core of the graph-theoretic VC-tendency:

• unbounded VC-dim $N(v)\in\mathcal{F}(G)$6 linear twin-width through split, bipartite, and co-bipartite obstructions;
• bounded VC-dim $N(v)\in\mathcal{F}(G)$7 (Biedl et al., 19 Jun 2026).

The proof strategy passes through a contraction mechanism based on partitions by distinct neighborhoods. Suppose $N(v)\in\mathcal{F}(G)$8 is partitioned into disjoint sets $N(v)\in\mathcal{F}(G)$9 such that $N(v)\cap S=A$0, $N(v)\cap S=A$1, and for each $N(v)\cap S=A$2, only $N(v)\cap S=A$3 vertices outside $N(v)\cap S=A$4 are mixed on $N(v)\cap S=A$5. Then

$N(v)\cap S=A$6

The partition itself is extracted via a combinatorial lemma based on Haussler’s packing bound: in a VC-dimension-$N(v)\cap S=A$7 graph on $N(v)\cap S=A$8 vertices, for any subset $N(v)\cap S=A$9 of size $G$0 one cannot have more than $G$1 vertices whose neighborhoods in $G$2 differ pairwise by at least $G$3. Iterating this produces blocks with controlled mixed sets and yields the exponent $G$4 (Biedl et al., 19 Jun 2026).

Interval graphs provide a sharper special case. If $G$5 is an $G$6-vertex interval graph, then

$G$7

There is also a lower-bound construction of $G$8-vertex interval graphs with twin-width in $G$9; concretely, for $vcdim(G)$0 the construction yields twin-width at least

$vcdim(G)$1

This leaves an explicit gap even at VC-dimension $vcdim(G)$2 (Biedl et al., 19 Jun 2026).

A common misconception is that bounded VC-dimension should force bounded twin-width. The results do not state this. They state that graphs of bounded VC-dimension can have unbounded twin-width, but not linear twin-width; the guaranteed behavior is sub-linear, not constant or uniformly bounded (Biedl et al., 19 Jun 2026).

3. Identifying codes: logarithmic versus polynomial size

For identifying codes, VC-tendency takes the form of a dichotomy theorem for hereditary classes. Let $vcdim(G)$3 be hereditary and write $vcdim(G)$4. Exactly one of the following holds. If $vcdim(G)$5, then for every integer $vcdim(G)$6 there is a graph $vcdim(G)$7 with

$vcdim(G)$8

If $vcdim(G)$9, then there is an exponent

$V$0

such that every twin-free $V$1 on $V$2 vertices satisfies

$V$3

Thus VC-dimension completely determines whether identifying codes can be logarithmic or must be polynomially large (Bousquet et al., 2014).

The finite-dimension side is an application of Sauer’s lemma. Any identifying code $V$4 in a twin-free graph must induce $V$5 distinct traces of the $V$6 closed neighborhoods on $V$7, so

$V$8

whence $V$9. The infinite-dimension side is witnessed by shattered sets: if a graph in $\mathcal{F}(G)=\{N(v):v\in V\}.$00 has a shattered set $\mathcal{F}(G)=\{N(v):v\in V\}.$01 of size $\mathcal{F}(G)=\{N(v):v\in V\}.$02, then there are $\mathcal{F}(G)=\{N(v):v\in V\}.$03 further vertices whose closed neighborhoods realize all nonempty subsets of $\mathcal{F}(G)=\{N(v):v\in V\}.$04, and one constructs an identifying code of size $\mathcal{F}(G)=\{N(v):v\in V\}.$05 (Bousquet et al., 2014).

This threshold behavior has direct algorithmic consequences. If $\mathcal{F}(G)=\{N(v):v\in V\}.$06, then $\mathcal{F}(G)=\{N(v):v\in V\}.$07 must contain, as induced subgraphs, all bipartite graphs or all split graphs or all co-bipartite graphs, and Min ID Code is log-APX-hard on $\mathcal{F}(G)=\{N(v):v\in V\}.$08. If $\mathcal{F}(G)=\{N(v):v\in V\}.$09 is finite, one does not obtain a uniform approximation theorem. Interval graphs, which have $\mathcal{F}(G)=\{N(v):v\in V\}.$10, admit a polynomial-time $\mathcal{F}(G)=\{N(v):v\in V\}.$11-approximation algorithm. In contrast, on $\mathcal{F}(G)=\{N(v):v\in V\}.$12-free bipartite graphs, also of VC-dimension $\mathcal{F}(G)=\{N(v):v\in V\}.$13, Min ID Code cannot be approximated within a factor $\mathcal{F}(G)=\{N(v):v\in V\}.$14 for some $\mathcal{F}(G)=\{N(v):v\in V\}.$15, unless $\mathcal{F}(G)=\{N(v):v\in V\}.$16 (Bousquet et al., 2014).

This establishes an important limitation of the VC-tendency viewpoint. Finite VC-dimension controls the asymptotic minimum size of identifying codes, but it does not by itself force constant-factor approximability. The interval-graph and $\mathcal{F}(G)=\{N(v):v\in V\}.$17-free bipartite cases demonstrate that identical VC-dimension can coexist with sharply different optimization complexity (Bousquet et al., 2014).

4. Learning-theoretic manifestations: GCNNs, GNNs, and WL color complexity

In deep learning, VC-tendency concerns how architectural parameters and combinatorial graph invariants govern capacity. For group convolutional neural networks, let $\mathcal{F}(G)=\{N(v):v\in V\}.$18 denote the class of GCNNs with kernel-basis dimension $\mathcal{F}(G)=\{N(v):v\in V\}.$19, layer widths $\mathcal{F}(G)=\{N(v):v\in V\}.$20, and discretization resolution $\mathcal{F}(G)=\{N(v):v\in V\}.$21. If

$\mathcal{F}(G)=\{N(v):v\in V\}.$22

then the layer-wise upper bound is

$\mathcal{F}(G)=\{N(v):v\in V\}.$23

For the class $\mathcal{F}(G)=\{N(v):v\in V\}.$24 of GCNNs with at most $\mathcal{F}(G)=\{N(v):v\in V\}.$25 layers, at most $\mathcal{F}(G)=\{N(v):v\in V\}.$26 weights, and resolution $\mathcal{F}(G)=\{N(v):v\in V\}.$27, the upper and lower bounds imply, up to constant factors,

$\mathcal{F}(G)=\{N(v):v\in V\}.$28

The $\mathcal{F}(G)=\{N(v):v\in V\}.$29 term matches the familiar scaling for fully connected ReLU networks, while the additional $\mathcal{F}(G)=\{N(v):v\in V\}.$30 term reflects the dependence on group discretization resolution (Sepliarskaia et al., 2024).

The same study makes the interpretation explicit: weight sharing does not reduce VC significantly compared to DNNs for fixed $\mathcal{F}(G)=\{N(v):v\in V\}.$31, but it controls the number of weights $\mathcal{F}(G)=\{N(v):v\in V\}.$32 itself via shared filters. Input resolution enters only logarithmically, so doubling $\mathcal{F}(G)=\{N(v):v\in V\}.$33 adds only $\mathcal{F}(G)=\{N(v):v\in V\}.$34 to VC. In the limit $\mathcal{F}(G)=\{N(v):v\in V\}.$35 for continuous groups, VC becomes infinite, matching known infinite VC for continuous two-layer GCNNs (Sepliarskaia et al., 2024).

For graph neural networks analyzed through Weisfeiler–Leman, the threshold is expressed in three regimes. In the unbounded-order regime, for bit-length $\mathcal{F}(G)=\{N(v):v\in V\}.$36 one has $\mathcal{F}(G)=\{N(v):v\in V\}.$37, and fixed-width, fixed-depth classes $\mathcal{F}(G)=\{N(v):v\in V\}.$38 have infinite VC dimension once arbitrarily high bit-length is allowed. In the bounded-order regime, if

$\mathcal{F}(G)=\{N(v):v\in V\}.$39

then

$\mathcal{F}(G)=\{N(v):v\in V\}.$40

On subclasses of bounded color complexity $\mathcal{F}(G)=\{N(v):v\in V\}.$41, a Bartlett-style bound yields, for piece-wise-linear activations,

$\mathcal{F}(G)=\{N(v):v\in V\}.$42

where $\mathcal{F}(G)=\{N(v):v\in V\}.$43. Thus VC grows only logarithmically in the $\mathcal{F}(G)=\{N(v):v\in V\}.$44-WL color complexity $\mathcal{F}(G)=\{N(v):v\in V\}.$45 (Morris et al., 2023).

Empirical results in the same work align with these theorems. Increasing feature dimension $\mathcal{F}(G)=\{N(v):v\in V\}.$46 at fixed depth enlarges the train–test gap, increasing the number of distinct $\mathcal{F}(G)=\{N(v):v\in V\}.$47-WL color histograms enlarges the gap until saturation, and increasing simulated bit-length improves the ability to memorize random labels on synthetic trees. The authors report that these observations confirm the predicted dependence of VC on parameter count, WL-distinguishable count, and bit precision (Morris et al., 2023).

A plausible implication is that, in neural architectures, VC-tendency is not a binary bounded-versus-unbounded statement but a scaling law. Capacity tracks parameter count, depth, discretization resolution, and WL distinguishability in quantitatively different ways, with logarithmic dependence on $\mathcal{F}(G)=\{N(v):v\in V\}.$48 and $\mathcal{F}(G)=\{N(v):v\in V\}.$49 but linear dependence on $\mathcal{F}(G)=\{N(v):v\in V\}.$50 and $\mathcal{F}(G)=\{N(v):v\in V\}.$51 up to logarithmic factors (Sepliarskaia et al., 2024, Morris et al., 2023).

5. VC density and dp-rank in model theory

In model theory, VC-tendency concerns the asymptotic growth of shatter counts as the number of parameters increases. For every complete NIP theory $\mathcal{F}(G)=\{N(v):v\in V\}.$52 and every $\mathcal{F}(G)=\{N(v):v\in V\}.$53,

$\mathcal{F}(G)=\{N(v):v\in V\}.$54

Here $\mathcal{F}(G)=\{N(v):v\in V\}.$55 is the supremum of VC densities of formulas with $\mathcal{F}(G)=\{N(v):v\in V\}.$56 parameter variables, and $\mathcal{F}(G)=\{N(v):v\in V\}.$57 is the maximum depth of an ICT pattern in $\mathcal{F}(G)=\{N(v):v\in V\}.$58 variables. Consequently, strong dependence is equivalent to finite VC density (Johnson, 2011).

The lower bound $\mathcal{F}(G)=\{N(v):v\in V\}.$59 is obtained by showing that an ICT-pattern of depth $\mathcal{F}(G)=\{N(v):v\in V\}.$60 forces $\mathcal{F}(G)=\{N(v):v\in V\}.$61 on suitable finite sets. The upper bound $\mathcal{F}(G)=\{N(v):v\in V\}.$62 is derived through a combinatorial alternative: if a formula has VC-density strictly greater than $\mathcal{F}(G)=\{N(v):v\in V\}.$63, then one can extract an ICT-pattern of depth $\mathcal{F}(G)=\{N(v):v\in V\}.$64. This is the point at which combinatorial growth translates into model-theoretic independence (Johnson, 2011).

Several standard theories illustrate the tendency. In o-minimal theories one has $\mathcal{F}(G)=\{N(v):v\in V\}.$65, hence $\mathcal{F}(G)=\{N(v):v\in V\}.$66 or $\mathcal{F}(G)=\{N(v):v\in V\}.$67, and more careful geometry yields $\mathcal{F}(G)=\{N(v):v\in V\}.$68. In algebraically closed fields, stability gives $\mathcal{F}(G)=\{N(v):v\in V\}.$69 for all $\mathcal{F}(G)=\{N(v):v\in V\}.$70, so $\mathcal{F}(G)=\{N(v):v\in V\}.$71. In $\mathcal{F}(G)=\{N(v):v\in V\}.$72-minimal or $\mathcal{F}(G)=\{N(v):v\in V\}.$73-adic settings, one has $\mathcal{F}(G)=\{N(v):v\in V\}.$74, hence $\mathcal{F}(G)=\{N(v):v\in V\}.$75, and more generally $\mathcal{F}(G)=\{N(v):v\in V\}.$76, $\mathcal{F}(G)=\{N(v):v\in V\}.$77 (Johnson, 2011).

The broader pattern is that $\mathcal{F}(G)=\{N(v):v\in V\}.$78 grows linearly in $\mathcal{F}(G)=\{N(v):v\in V\}.$79 up to an additive constant of $\mathcal{F}(G)=\{N(v):v\in V\}.$80 in natural examples. The main unresolved issue recorded there is whether the upper bound can be sharpened to

$\mathcal{F}(G)=\{N(v):v\in V\}.$81

which would imply integrality of VC-density and exact coincidence with dp-rank. No counter-example is known, but the general proof requires the additional “$\mathcal{F}(G)=\{N(v):v\in V\}.$82” slack (Johnson, 2011).

6. Algorithmic thresholds, limitations, and open problems

For Hitting Set, low VC-dimension yields a sharp but limited tractability picture. If either the primal or dual VC-dimension is $\mathcal{F}(G)=\{N(v):v\in V\}.$83, then Hitting Set is solvable in polynomial time. Once the VC-dimension rises to $\mathcal{F}(G)=\{N(v):v\in V\}.$84, even with dual VC-dimension also $\mathcal{F}(G)=\{N(v):v\in V\}.$85, the parameterized problem becomes W[1]-hard, and under ETH there is no algorithm of time $\mathcal{F}(G)=\{N(v):v\in V\}.$86 (Bringmann et al., 2015).

The same work shows that the raw VC-dimension can be too coarse, and introduces a finer threshold via $\mathcal{F}(G)=\{N(v):v\in V\}.$87-systems. Hitting Set on any $\mathcal{F}(G)=\{N(v):v\in V\}.$88-system is solvable in polynomial time, while Hitting Set on $\mathcal{F}(G)=\{N(v):v\in V\}.$89-systems is NP-hard. Thus, for $\mathcal{F}(G)=\{N(v):v\in V\}.$90, there is a sharp threshold:

• $\mathcal{F}(G)=\{N(v):v\in V\}.$91,
• $\mathcal{F}(G)=\{N(v):v\in V\}.$92 NP-hard (Bringmann et al., 2015).

These results guard against an overly strong reading of VC-tendency. Bounded VC-dimension often predicts improved behavior, but it does not uniformly imply easy optimization, fixed-parameter tractability, or bounded structural width. Three examples from the cited literature make this explicit. First, bounded VC-dimension does not imply bounded twin-width; it implies sub-linear twin-width growth (Biedl et al., 19 Jun 2026). Second, finite VC-dimension does not imply constant-factor approximability for identifying codes; interval graphs and $\mathcal{F}(G)=\{N(v):v\in V\}.$93-free bipartite graphs already diverge at VC-dimension $\mathcal{F}(G)=\{N(v):v\in V\}.$94 (Bousquet et al., 2014). Third, low VC-dimension does not imply parameterized tractability for Hitting Set; W[1]-hardness already appears at VC-dimension $\mathcal{F}(G)=\{N(v):v\in V\}.$95 with dual VC-dimension $\mathcal{F}(G)=\{N(v):v\in V\}.$96 (Bringmann et al., 2015).

Several open problems remain central. For twin-width, it is unknown whether the exponent $\mathcal{F}(G)=\{N(v):v\in V\}.$97 is optimal for general VC-dimension $\mathcal{F}(G)=\{N(v):v\in V\}.$98, and even for interval graphs there is a gap between the $\mathcal{F}(G)=\{N(v):v\in V\}.$99 upper bound and the $S\subseteq V$00 lower bound. For $S\subseteq V$01, VC-dimension-$S\subseteq V$02 graphs are disjoint unions of cliques and have twin-width $S\subseteq V$03; for $S\subseteq V$04, the true exponent may lie strictly between $S\subseteq V$05 and $S\subseteq V$06 (Biedl et al., 19 Jun 2026). For model theory, the status of the “$S\subseteq V$07” in $S\subseteq V$08 remains open (Johnson, 2011). For Hitting Set, open questions include whether the problem is FPT on $S\subseteq V$09-systems and whether one can fully characterize the classical and parameterized complexity for general $S\subseteq V$10-systems when $S\subseteq V$11 (Bringmann et al., 2015).

Taken together, these works show that VC-tendency is best understood as a family of threshold principles. The precise threshold variable may be VC-dimension, VC density, dual VC-dimension, WL color complexity, bit-length, or a refined shatter parameter, but the recurring pattern is that combinatorial shattering complexity governs transitions in width, code size, learnability, expressivity, and algorithmic hardness (Biedl et al., 19 Jun 2026, Bousquet et al., 2014, Sepliarskaia et al., 2024, Morris et al., 2023, Johnson, 2011, Bringmann et al., 2015).