---
title: 'Shrub-depth: Dense Graph Parameter'
url: https://www.emergentmind.com/topics/shrub-depth
type: topic
---

# Shrub-depth: Dense Graph Parameter

Shrub-depth is a graph parameter for graph classes that captures a notion of bounded height in dense graphs. It is defined through bounded-height tree-models in which adjacency is determined by leaf colours and the depth of the least common ancestor, and it was introduced as a dense analogue of tree-depth that remains well behaved under graph complementation and induced subgraphs [1707.00359]. A central structural fact is that bounded shrub-depth is equivalent to bounded rank-depth, the branch-depth of the cut-rank function, so shrub-depth can be studied either through coloured rooted trees or through \(\mathbb F_2\)-cut decompositions [1903.11988].

## 1. Tree-models and the formal definition

A **tree-model of \(m\) colours and depth \(d\)**, or a \((d,m)\)-tree-model, for a graph \(G\) is a pair \((T,S)\) where \(T\) is a rooted tree of height \(d\), every root-to-leaf path has length exactly \(d\), the leaves of \(T\) are exactly the vertices \(V(G)\), each leaf is assigned one of the colours in \([m]\), and \(S \subseteq [m]^2 \times [d]\) is a symmetric signature. For vertices \(u,v\) of colours \(i,j\), if their distance in \(T\) is \(2\ell\), then
\[
\{u,v\}\in E(G) \iff (i,j,\ell)\in S.
\]
Equivalently, adjacency depends only on the colours of the leaves and the depth of their least common ancestor [2506.22269].

A graph class \(\mathcal G\) has **shrub-depth at most \(d\)** if there exists some \(m\) such that every graph \(G\in\mathcal G\) admits a \((d,m)\)-tree-model. In the notation of the original review, \(\TM dm\) denotes the class of graphs having a tree-model with depth \(d\) and \(m\) colours, and a class has shrub-depth \(d\) if it is contained in some \(\TM dm\) but in no \(\TM{d-1}{m'}\) for any \(m'\) [1707.00359].

The parameter is inherently a parameter of classes rather than of single finite graphs; the 2014 vertex-minor paper explicitly notes that shrub-depth is primarily a notion for graph classes, not for a single graph [1403.7024]. The classes \(\TM dm\) are closed under complements and induced subgraphs, which was one of the explicit design criteria behind the notion, but they are not closed under arbitrary subgraphs [1707.00359].

Several basic examples are standard. Complete graphs satisfy \(K_n\in \TM 1 1\), complete bipartite graphs satisfy \(K_{n,n}\in \TM 1 2\), and \(\TM 1m\) is exactly the class of graphs of neighbourhood diversity at most \(m\) [1707.00359].

## 2. Equivalent formulations and placement among depth parameters

Shrub-depth was designed to parallel the relation between tree-depth and tree-width in the dense setting. If a class has tree-depth at most \(d\), then it has shrub-depth at most \(d\). Conversely, bounded shrub-depth implies bounded linear clique-width and also bounded linear rank-width, but the converses fail in general: complete graphs witness failure of the first converse, while paths witness failure of the second [1707.00359], [2507.12697].

A second formulation is **SC-depth**. One starts from \(\SC0=\{K_1\}\), and if \(G_1,\dots,G_p\in \SC k\) and \(H=G_1\dot\cup \cdots \dot\cup G_p\), then for every \(X\subseteq V(H)\) the graph \(\overline{H}^{\,X}\), obtained by complementing the edges inside \(X\), belongs to \(\SC{k+1}\). Bounded SC-depth is equivalent to bounded shrub-depth [1707.00359], [2501.13903].

A third formulation is **rank-depth**. For a graph \(G\) and a set \(S\subseteq V(G)\), let
\[
\rho_G(S)=\operatorname{rank}_{\mathbb F_2}A(G)[S,V(G)\setminus S].
\]
A decomposition \((T,\sigma)\) of \(G\) is a tree whose leaves are bijectively identified with \(V(G)\). For each internal node \(v\), the components of \(T-v\) induce a partition \(\mathcal P_v\) of \(V(G)\), and the width of \(v\) is
\[
\max_{\mathcal P'\subseteq \mathcal P_v}\rho_G\!\left(\bigcup_{X\in\mathcal P'}X\right).
\]
The rank-depth of \(G\) is the minimum \(k\) such that \(G\) admits a \((k,k)\)-decomposition. DeVos, Kwon, and Oum proved that a graph class has bounded rank-depth if and only if it has bounded shrub-depth [2507.12697].

| Formulation | Core object | Boundedness statement |
|---|---|---|
| Tree-models | Rooted tree of depth \(d\) with \(m\) leaf colours and signature \(S\) | Defines bounded shrub-depth |
| SC-depth | Iterated disjoint unions and set complementations | Bounded SC-depth iff bounded shrub-depth |
| Rank-depth | Branch-depth of the cut-rank function \(\rho_G\) | Bounded rank-depth iff bounded shrub-depth |

This equivalence places shrub-depth inside the general branch-depth program for connectivity functions: tree-depth corresponds to branch-depth of the usual edge-connectivity function, whereas shrub-depth corresponds to branch-depth of cut-rank [1903.11988].

## 3. Obstructions and large-depth structure

The first major obstruction theorem identified paths as the fundamental vertex-minor obstruction. For every positive integer \(t\), there exists an integer \(N(t)\) such that every graph of rank-depth at least \(N(t)\) contains a vertex-minor isomorphic to the path \(P_t\). Since bounded rank-depth is equivalent to bounded shrub-depth, it follows that for every integer \(t\), the class of graphs with no vertex-minor isomorphic to \(P_t\) has bounded shrub-depth [1911.00230].

For hereditary classes, the obstruction theory is now much more explicit. The 2025 paper "Forbidden Induced Subgraphs for Bounded Shrub-Depth and the Expressive Power of MSO" proves that for every hereditary graph class \(C\), bounded shrub-depth is equivalent to excluding all flipped half-graphs of some order \(t\) and all flipped \(tP_t\) as induced subgraphs; the proofs also show that \(tP_t\) can be replaced by \(3P_t\). Here the half-graph \(H_t\) has bipartition \(\{a_1,\dots,a_t\}\cup \{b_1,\dots,b_t\}\) with
\[
a_i b_j\in E(H_t)\iff i<j,
\]
and a flip \(H=G\oplus(P,F)\) toggles adjacency between parts of a partition \(P\) according to a symmetric relation \(F\subseteq P^2\) [2501.13903].

The hereditary theory also clarifies that the obstruction family is genuinely two-sided. The paper notes that the \(2P_t\) version remains open and the \(1P_t\) version is false, since every graph on \(t\) vertices is a flipped \(1P_t\) [2501.13903]. This sharpens earlier results which only yielded, for each fixed shrub-depth bound, the existence of some finite forbidden induced subgraph set [1707.00359].

Within bipartite permutation graphs, the obstruction picture becomes especially rigid. Lozin, Razgon, and Zamaraev proved that the classes of **chain graphs** and **linear forests** are the only two minimal hereditary subclasses of bipartite permutation graphs of unbounded shrub-depth and rank-depth. Chain graphs are precisely the \(2K_2\)-free bipartite graphs, while linear forests are graphs in which every connected component is a path \(P_k\) [2010.14467].

## 4. Vertex-minors, pivot-minors, and transfer from tree-depth

Vertex-minors are obtained by sequences of local complementations followed by taking induced subgraphs. Shrub-depth is monotone under taking vertex-minors: if a class has bounded shrub-depth, then every class of its vertex-minors has bounded shrub-depth as well [1403.7024]. This monotonicity is mirrored on the rank-depth side by the fact that rank-depth does not increase under vertex-minors [1911.00230].

The same 2014 study established a structural transfer theorem from tree-depth: for any class \(\mathcal S\) of bounded shrub-depth, there exists an integer \(d\) such that every graph in \(\mathcal S\) is a vertex-minor of some graph of tree-depth \(d\). In this sense, bounded shrub-depth classes are exactly the graph classes obtainable from bounded tree-depth classes through the operations defining vertex-minors [1403.7024].

Pivot-minors are more restrictive. Without further hypotheses, bounded shrub-depth classes are not all pivot-minor closures of bounded tree-depth classes: if \(t>3^{d-1}\), then no graph of tree-depth at most \(d\) has \(K_t\) as a pivot-minor, while complete graphs have shrub-depth \(1\) [1403.7024]. For bipartite graphs, however, the pivot-minor version does hold: for any class \(\mathcal S\) of bounded shrub-depth, there exists \(d\) such that every bipartite \(G\in\mathcal S\) is a pivot-minor of a graph of tree-depth \(d\) [1403.7024].

A sharper pivot-minor obstruction theorem was obtained by Kim, Kwon, Oum, and their coauthor in 2025. There exists a function \(f:\mathbb N\to\mathbb N\) such that for every \(t\), every graph of rank-depth at least \(f(t)\) has a pivot-minor isomorphic to either \(P_t\) or \(K_tK_t\), where \(K_tK_t\) is formed from two disjoint copies of \(K_t\) by adding the half graph between them: if the two cliques are ordered \(u_1,\dots,u_t\) and \(v_1,\dots,v_t\), then \(u_i\) is adjacent to \(v_j\) exactly when \(i\ge j\). This answers the open problem posed by Kwon, McCarty, Oum, and Wollan in 2021 [2507.12697].

The proof passes through induced-subgraph obstructions due to Mählmann: sufficiently large rank-depth forces an induced subgraph isomorphic to one of
\[
K_sK_s,\qquad K_s\overline{K_s},\qquad \overline{K_s}\overline{K_s},\qquad \text{or a flipped } sP_s,
\]
and each of these contains one of the unavoidable pivot-minors \(P_t\) or \(K_tK_t\) [2507.12697]. Because bounded rank-depth and bounded shrub-depth are equivalent, the theorem also describes the unavoidable pivot-minor patterns in classes of large shrub-depth.

## 5. Logical and model-theoretic characterizations

Shrub-depth has an exact interpretation-theoretic characterization. A class of graphs has shrub-depth at most \(d\) if and only if it has a simple interpretation in a class of finite coloured rooted trees of height at most \(d\) [1707.00359]. This result is one of the main reasons the parameter is so effective in finite model theory: bounded shrub-depth is precisely the graph-theoretic content of bounded-height tree interpretations.

The same review shows that shrub-depth captures the lower finite levels of the MSO\(_1\) and CMSO\(_1\) transduction hierarchies. For any graph class \(\mathcal G\) of bounded shrub-depth there is an integer \(d\) such that
\[
\mathcal G \equiv \mathcal T_d
\]
in the scope of MSO\(_1\) or CMSO\(_1\) transductions, where \(\mathcal T_d\) denotes rooted trees of height at most \(d\) [1707.00359]. Related transduction results show that structurally bounded treedepth is the same as bounded shrubdepth, and that the corresponding encodings can be phrased as bounded-diameter shrub-decompositions whose hidden graph has bounded treedepth [2104.10446].

For arbitrary finite or infinite graphs admitting fixed tree-model parameters, the model theory is unusually tame. The class \(\mathbf{TM}_r(d)\) is closed under ultraproducts and ultraroots, hence elementary, and every graph in \(\mathbf{TM}_r(d)\) is MSO-pseudo-finite relative to the finite graphs of \(\mathbf{TM}_r(d)\). The index of the MSO\([m]\)-equivalence relation on \(\mathbf{TM}_r(d)\) is bounded by a \((d+1)\)-fold exponential in \(m\) [2202.06308]. A parallel development for \(\mathrm{TM}_{r,p}(d)\) establishes an extended Löwenheim-Skolem property with elementary witness functions, pseudo-finiteness, a compactness theorem over the class, and the collapse \(\mathrm{MSO}=\mathrm{FO}\) over \(\mathrm{TM}_{r,p}(d)\) and hence over bounded shrub-depth classes [2010.05799].

For hereditary graph classes, the logical boundary is now exact. Bounded shrub-depth is equivalent to MSO-stability, to monadic MSO-stability, to CMSO-stability, to not \(1\)-dimensionally FO-interpret the class of all paths, and to FO and MSO having the same expressive power on the class [2501.13903]. In particular, the 2025 forbidden-subgraph paper confirms the conjecture of Gajarský and Hliněný by proving the converse to the earlier FO\(=\)MSO result: on every hereditary class of unbounded shrub-depth, MSO is more expressive than FO [2501.13903].

## 6. Algorithmic and extremal consequences

Shrub-depth is algorithmically relevant because bounded depth changes the space complexity of decomposition-based dynamic programming. On an \(n\)-vertex graph equipped with a \((d,k)\)-tree-model, the 2023 paper "Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth" gives an algorithm for **Independent Set** running in time
\[
2^{O(dk)}\cdot n^{O(1)}
\]
using
\[
O(dk^2\log n)
\]
space, an algorithm for **Max Cut** running in time
\[
n^{O(dk)}
\]
using
\[
O(dk\log n)
\]
space, and a randomized algorithm for **Dominating Set** running in time
\[
2^{O(dk)}\cdot n^{O(1)}
\]
using polynomial space, correct with probability at least \(1/2\) [2307.01285].

The same work proves a conditional lower bound for polynomial-space algorithms on low shrub-depth graphs. Under the stated assumption on the complexity of **Longest Common Subsequence**, there is no algorithm solving **Independent Set** on graphs supplied with \((d,k)\)-tree-models satisfying \(d\le \delta(k)\) for an unbounded computable function \(\delta\), in time
\[
2^{O(k)}\cdot n^{O(1)}
\]
and space \(n^{O(1)}\). Thus the dependence on the depth parameter in the exponent is not merely an artefact of the upper bounds [2307.01285].

Shrub-depth also controls extremal density in traceable graphs. If a traceable graph \(G\) on \(n\) vertices admits a \((d,m)\)-tree-model, then
\[
|E(G)| \ge \frac{1}{16m}\cdot n^{1+\frac{1}{2^d-1}},
\]
and for any fixed depth \(d\) there are infinitely many \(n\) for which there exists a traceable graph \(G\) admitting a \((d,2^{d-1})\)-tree-model with
\[
|E(G)| \le 2^{O(d)}\cdot n^{1+\frac{1}{2^d-1}}.
\]
Accordingly, for fixed \(d\), the edge density of traceable graphs of bounded shrub-depth is asymptotically
\[
\Theta\!\left(n^{1+\frac{1}{2^d-1}}\right),
\]
which the paper describes as **slightly superlinear** [2506.22269].

These developments collectively position shrub-depth as a dense depth parameter with unusually strong structural, logical, and algorithmic regularity: it admits equivalent formulations through tree-models, SC-depth, and rank-depth; it has explicit obstruction theories under induced subgraphs, vertex-minors, and pivot-minors; and it marks an exact boundary for FO versus MSO on hereditary graph classes [1903.11988], [2501.13903].

Source: https://www.emergentmind.com/topics/shrub-depth