Shrub-depth: Dense Graph Parameter

• Shrub-depth is a graph parameter that quantifies bounded tree-model depth in dense graphs by determining adjacency via leaf colours and least common ancestor depths.
• It is equivalent to both SC-depth and rank-depth, establishing key connections with tree-depth and ensuring closure under complementation and induced subgraphs.
• It has significant algorithmic implications, enabling space-efficient dynamic programming for problems like Independent Set, Max Cut, and Dominating Set on bounded shrub-depth graphs.

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 (Ganian et al., 2017). 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 (DeVos et al., 2019).

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 $m$0, each leaf is assigned one of the colours in $m$1, and $m$2 is a symmetric signature. For vertices $m$3 of colours $m$4, if their distance in $m$5 is $m$6, then

$m$7

Equivalently, adjacency depends only on the colours of the leaves and the depth of their least common ancestor (Dvořák et al., 27 Jun 2025).

A graph class $m$8 has shrub-depth at most $m$9 if there exists some $d$0 such that every graph $d$1 admits a $d$2-tree-model. In the notation of the original review, $d$3 denotes the class of graphs having a tree-model with depth $d$4 and $d$5 colours, and a class has shrub-depth $d$6 if it is contained in some $d$7 but in no $d$8 for any $d$9 (Ganian et al., 2017).

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 (Hliněný et al., 2014). The classes $(d,m)$0 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 (Ganian et al., 2017).

Several basic examples are standard. Complete graphs satisfy $(d,m)$1, complete bipartite graphs satisfy $(d,m)$2, and $(d,m)$3 is exactly the class of graphs of neighbourhood diversity at most $(d,m)$4 (Ganian et al., 2017).

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,m)$5, then it has shrub-depth at most $(d,m)$6. 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 (Ganian et al., 2017, Ahn et al., 17 Jul 2025).

A second formulation is SC-depth. One starts from $(d,m)$7, and if $(d,m)$8 and $(d,m)$9, then for every $G$0 the graph $G$1, obtained by complementing the edges inside $G$2, belongs to $G$3. Bounded SC-depth is equivalent to bounded shrub-depth (Ganian et al., 2017, Mählmann, 23 Jan 2025).

A third formulation is rank-depth. For a graph $G$4 and a set $G$5, let

$G$6

A decomposition $G$7 of $G$8 is a tree whose leaves are bijectively identified with $G$9. For each internal node $(T,S)$0, the components of $(T,S)$1 induce a partition $(T,S)$2 of $(T,S)$3, and the width of $(T,S)$4 is

$(T,S)$5

The rank-depth of $(T,S)$6 is the minimum $(T,S)$7 such that $(T,S)$8 admits a $(T,S)$9-decomposition. DeVos, Kwon, and Oum proved that a graph class has bounded rank-depth if and only if it has bounded shrub-depth (Ahn et al., 17 Jul 2025).

Formulation	Core object	Boundedness statement
Tree-models	Rooted tree of depth $T$0 with $T$1 leaf colours and signature $T$2	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 $T$3	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 (DeVos et al., 2019).

3. Obstructions and large-depth structure

The first major obstruction theorem identified paths as the fundamental vertex-minor obstruction. For every positive integer $T$4, there exists an integer $T$5 such that every graph of rank-depth at least $T$6 contains a vertex-minor isomorphic to the path $T$7. Since bounded rank-depth is equivalent to bounded shrub-depth, it follows that for every integer $T$8, the class of graphs with no vertex-minor isomorphic to $T$9 has bounded shrub-depth (Kwon et al., 2019).

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 $d$0, bounded shrub-depth is equivalent to excluding all flipped half-graphs of some order $d$1 and all flipped $d$2 as induced subgraphs; the proofs also show that $d$3 can be replaced by $d$4. Here the half-graph $d$5 has bipartition $d$6 with

$d$7

and a flip $d$8 toggles adjacency between parts of a partition $d$9 according to a symmetric relation $d$0 (Mählmann, 23 Jan 2025).

The hereditary theory also clarifies that the obstruction family is genuinely two-sided. The paper notes that the $d$1 version remains open and the $d$2 version is false, since every graph on $d$3 vertices is a flipped $d$4 (Mählmann, 23 Jan 2025). This sharpens earlier results which only yielded, for each fixed shrub-depth bound, the existence of some finite forbidden induced subgraph set (Ganian et al., 2017).

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 $d$5-free bipartite graphs, while linear forests are graphs in which every connected component is a path $d$6 (Alecu et al., 2020).

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 (Hliněný et al., 2014). This monotonicity is mirrored on the rank-depth side by the fact that rank-depth does not increase under vertex-minors (Kwon et al., 2019).

The same 2014 study established a structural transfer theorem from tree-depth: for any class $d$7 of bounded shrub-depth, there exists an integer $d$8 such that every graph in $d$9 is a vertex-minor of some graph of tree-depth $T$0. In this sense, bounded shrub-depth classes are exactly the graph classes obtainable from bounded tree-depth classes through the operations defining vertex-minors (Hliněný et al., 2014).

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$1, then no graph of tree-depth at most $T$2 has $T$3 as a pivot-minor, while complete graphs have shrub-depth $T$4 (Hliněný et al., 2014). For bipartite graphs, however, the pivot-minor version does hold: for any class $T$5 of bounded shrub-depth, there exists $T$6 such that every bipartite $T$7 is a pivot-minor of a graph of tree-depth $T$8 (Hliněný et al., 2014).

A sharper pivot-minor obstruction theorem was obtained by Kim, Kwon, Oum, and their coauthor in 2025. There exists a function $T$9 such that for every $m$00, every graph of rank-depth at least $m$01 has a pivot-minor isomorphic to either $m$02 or $m$03, where $m$04 is formed from two disjoint copies of $m$05 by adding the half graph between them: if the two cliques are ordered $m$06 and $m$07, then $m$08 is adjacent to $m$09 exactly when $m$10. This answers the open problem posed by Kwon, McCarty, Oum, and Wollan in 2021 (Ahn et al., 17 Jul 2025).

The proof passes through induced-subgraph obstructions due to Mählmann: sufficiently large rank-depth forces an induced subgraph isomorphic to one of

$m$11

and each of these contains one of the unavoidable pivot-minors $m$12 or $m$13 (Ahn et al., 17 Jul 2025). 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 $m$14 if and only if it has a simple interpretation in a class of finite coloured rooted trees of height at most $m$15 (Ganian et al., 2017). 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$m$16 and CMSO$m$17 transduction hierarchies. For any graph class $m$18 of bounded shrub-depth there is an integer $m$19 such that

$m$20

in the scope of MSO$m$21 or CMSO$m$22 transductions, where $m$23 denotes rooted trees of height at most $m$24 (Ganian et al., 2017). 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 (Dreier, 2021).

For arbitrary finite or infinite graphs admitting fixed tree-model parameters, the model theory is unusually tame. The class $m$25 is closed under ultraproducts and ultraroots, hence elementary, and every graph in $m$26 is MSO-pseudo-finite relative to the finite graphs of $m$27. The index of the MSO$m$28-equivalence relation on $m$29 is bounded by a $m$30-fold exponential in $m$31 (Sankaran, 2022). A parallel development for $m$32 establishes an extended Löwenheim-Skolem property with elementary witness functions, pseudo-finiteness, a compactness theorem over the class, and the collapse $m$33 over $m$34 and hence over bounded shrub-depth classes (Sankaran, 2020).

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 $m$35-dimensionally FO-interpret the class of all paths, and to FO and MSO having the same expressive power on the class (Mählmann, 23 Jan 2025). In particular, the 2025 forbidden-subgraph paper confirms the conjecture of Gajarský and Hliněný by proving the converse to the earlier FO$m$36MSO result: on every hereditary class of unbounded shrub-depth, MSO is more expressive than FO (Mählmann, 23 Jan 2025).

6. Algorithmic and extremal consequences

Shrub-depth is algorithmically relevant because bounded depth changes the space complexity of decomposition-based dynamic programming. On an $m$37-vertex graph equipped with a $m$38-tree-model, the 2023 paper "Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth" gives an algorithm for Independent Set running in time

$m$39

using

$m$40

space, an algorithm for Max Cut running in time

$m$41

using

$m$42

space, and a randomized algorithm for Dominating Set running in time

$m$43

using polynomial space, correct with probability at least $m$44 (Bergougnoux et al., 2023).

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 $m$45-tree-models satisfying $m$46 for an unbounded computable function $m$47, in time

$m$48

and space $m$49. Thus the dependence on the depth parameter in the exponent is not merely an artefact of the upper bounds (Bergougnoux et al., 2023).

Shrub-depth also controls extremal density in traceable graphs. If a traceable graph $m$50 on $m$51 vertices admits a $m$52-tree-model, then

$m$53

and for any fixed depth $m$54 there are infinitely many $m$55 for which there exists a traceable graph $m$56 admitting a $m$57-tree-model with

$m$58

Accordingly, for fixed $m$59, the edge density of traceable graphs of bounded shrub-depth is asymptotically

$m$60

which the paper describes as slightly superlinear (Dvořák et al., 27 Jun 2025).

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 (DeVos et al., 2019, Mählmann, 23 Jan 2025).