The Twin-Width of Graphs of Bounded VC-Dimension

Abstract: In this paper, we investigate which hereditary classes of graphs admit sub-linear (in the number of vertices) bounds on twin-width. By modifying conference graphs, we can show that split, bipartite, and co-bipartite graphs can all have linear twin-width. However, excluding an induced subgraph of each of these three types is equivalent to the class of graphs having bounded VC-dimension, as shown by Bousquet, Lagoutte, Li, Parreau and Thomassé. Graphs of bounded VC-dimension can have unbounded twin-width, but whether it can be linear was an open question. In this paper, we first present a tool for obtaining twin-width bounds in general by contracting a graph based on a partition by distinct neighbourhoods. Then, using this tool, we prove that graphs with bounded VC-dimension have twin-width at most sub-linear. We also obtain a separate, tighter upper bound for the class of interval graphs, as well as a lower bound.

• The paper demonstrates that hereditary graph classes with bounded VC-dimension have sublinear twin-width, leveraging novel contraction sequences.
• The introduced contraction tool partitions vertex neighborhoods to convert VC-dimension constraints into effective twin-width upper bounds.
• Explicit bounds for interval graphs highlight tightness of the results, contrasting with linear twin-width in split, bipartite, and co-bipartite graphs.

Twin-Width in Graphs of Bounded VC-Dimension

Introduction and Context

This paper rigorously investigates the landscape of twin-width within hereditary graph classes with bounded VC-dimension, bridging two major structural graph complexity parameters. Twin-width, a recent and increasingly influential invariant, generalizes both tree-width and clique-width, and is particularly potent for FO model checking, coloring, and enumeration problems. Bounded VC-dimension, rooted in learning theory, formalizes the complexity of families of vertex neighborhoods within a graph. The historically open question addressed is whether bounded VC-dimension guarantees sub-linear twin-width—a structural property with deep algorithmic and combinatorial consequences.

Preliminaries

For a graph $G$, let $V(G)$ and $E(G)$ denote its vertex and edge sets, and $N(v)$ the neighborhood of vertex $v$. The twin-width of $G$, denoted $\mathrm{tww}(G)$, is the minimum integer $d$ such that there exists an elimination (contraction) sequence of $G$ into a singleton where, at all steps, the maximum red degree does not exceed $d$. Red edges record disagreements during vertex contraction, serving as error tracking for the contraction process.

The VC-dimension $V(G)$0 is the size of the largest $V(G)$1 such that for all $V(G)$2, there exists $V(G)$3 with $V(G)$4. Computing VC-dimension is LOGNP-Complete, but its value provides strong combinatorial handles on the complexity of neighborhood systems.

The Twin-Width Contraction Tool

A key technical contribution is the contraction tool (Theorem~\ref{thm:secret-tww}), which leverages partitions based on neighborhood similarity. Given a partition $V(G)$5 where each $V(G)$6 (for $V(G)$7) is only mixed by few vertices in subsequent parts, the tool constructs a contraction sequence whose width is bounded in terms of $V(G)$8 (the number of sets), $V(G)$9 (the maximum number of vertices mixing any $E(G)$0), and $E(G)$1 (the size of the exceptional set $E(G)$2). This framework allows for sharp upper bounds once appropriate partitions are constructed—enabling the translation of structural set system bounds on $E(G)$3 into effective contraction sequences.

Main Results: Sub-linear Twin-Width for Bounded VC-Dimension

Upper Bound Construction

Using the twin-width contraction tool, the authors establish that for any $E(G)$4, $E(G)$5-vertex graphs with $E(G)$6 exhibit $E(G)$7. The proof recursively partitions the vertex set using VC-dimension stability properties (via Lemma~\ref{lm:vc-tww-2}), each time isolating a set of "similar" neighborhoods of size $E(G)$8 mixed by at most $E(G)$9 vertices, with $N(v)$0. This partition feeds into the contraction tool to yield the desired sublinear bound. Importantly, the result is tight up to polynomial factors—any hereditary class with unbounded VC-dimension admits arbitrarily large graphs with linear twin-width, as demonstrated by the constructed split, bipartite, and co-bipartite examples.

Tightness and Characterization

The analysis shows that hereditary classes of graphs admit sub-linear twin-width if and only if they have bounded VC-dimension. Notably, interval graphs (VC-dimension at most 2) serve as critical test cases for sharpness, as their class has super-polynomial size but unbounded twin-width.

Explicit Bounds for Interval Graphs

Interval graphs, a prototypical class within bounded VC-dimension, are analyzed using more refined partitions exploiting their interval structure. The authors partition the vertices according to right endpoints into $N(v)$1 blocks, each consisting of intervals whose right endpoints fall within a given range. For each block, only $N(v)$2 intervals from later blocks can be mixed on it (since the interval endpoints are distinct). This partition enables invocation of the contraction tool to obtain an explicit upper bound:

$N(v)$3

for all $N(v)$4-vertex interval graphs. This improves the exponent relative to the $N(v)$5 generic VC-dimension bound.

Figure 1: The constructed graph for Theorem~\ref{thm:tww-interval-lb}, exhibiting the separation between clique and stable set structure underpinning the interval graph lower bound construction.

Lower Bound

A complementary lower bound is established via interval graphs that are also split graphs. For $N(v)$6, a construction yields interval graphs with twin-width at least $N(v)$7, as contracting sets quickly forces high red degree or the presence of many neighborhood differences. These constructions tightly sandwich the true order of magnitude for the twin-width of interval graphs.

Contrasts with Split, Bipartite, and Co-bipartite Graphs

Augmenting prior observations, the authors prove that split, bipartite, and co-bipartite graphs can have linear twin-width. This is demonstrated by tensoring conference graphs—known to have linear twin-width—with $N(v)$8, ensuring the resultant graphs are bipartite while preserving high red degree upon contractions. Via the self-complementary nature of twin-width, this also holds for co-bipartite graphs, and by appropriate construction, for split graphs as well. This illustrates the precision of the VC-dimension/twin-width interface.

Numerical Bounds and Comparative Table

Specialized bounds for a variety of graph classes follow immediately:

Graph Class	VC-dim	Twin-width Upper Bound
Interval	≤2	$N(v)$9
Unit disk, permutation	≤3	$v$0
Disk, line, planar	≤4	$v$1

These express how improved contraction bounds arise for small-$v$2 classes, and situate the twin-width/VC-dimension hierarchy.

Implications and Prospects

The implications are twofold:

1. Algorithmic: Many FO-expressible problems are FPT on classes of bounded twin-width, and this work extends such tractability to all hereditary classes of bounded VC-dimension, quantifying the parameter's growth rate.
2. Structural: The paper provides a robust dichotomy theorem delineating hereditary classes of small twin-width precisely as those with bounded VC-dimension, independent of more familiar sparsity/density distinctions.

Extending the twin-width framework to broader notions of neighborhood complexity or alternative set system invariants is a promising avenue. Given the sharp lower and upper bounds, further refinement would require fundamentally new contraction techniques or combinatorial insights, possibly revealing new width parameters intermediate between VC-dimension and twin-width.

Conclusion

This work rigorously characterizes the asymptotic twin-width of hereditary classes via VC-dimension, establishing both existential and constructive bounds, and providing explicit upper and lower bounds in the prototypical case of interval graphs. The contraction tool developed generalizes to other structural decompositions and may serve as a template for bounding twin-width in other combinatorial and geometric graph classes. The results precisely link set system complexity (VC-dimension) with contraction-based width parameters, unifying disparate perspectives in combinatorics and algorithmic graph theory.