Rank-depth in Graph Theory

• Rank-depth is a depth-analogue of rank-width and a dense analogue of tree-depth defined through the branch-depth of the cut-rank function using radius-limited tree decompositions.
• It is equivalent to bounded shrub-depth and is monotone under vertex-minors, establishing its role in classifying dense graph classes through local complementation.
• The parameter leads to vertex-minor obstruction theorems, involving path obstructions and pivot-minor configurations like the half-graph clique structure K_tK_t.

Rank-depth is the branch-depth of the cut-rank function of a simple graph, and is therefore a depth-analogue of rank-width as well as a dense analogue of tree-depth. It is defined by tree decompositions of bounded radius whose cuts are measured over $\mathbb F_2$, and its boundedness is equivalent to bounded shrub-depth. The parameter is monotone under vertex-minors because cut-rank is invariant under local complementation, and large rank-depth is characterized by unavoidable path obstructions; in the pivot-minor setting, the corresponding obstruction theory also involves the graph $K_tK_t$, consisting of two disjoint cliques of size $t$ joined by a half graph (DeVos et al., 2019, Kwon et al., 2019, Ahn et al., 17 Jul 2025).

1. Definition through cut-rank and branch-depth

For a simple graph $G=(V,E)$ and a subset $X\subseteq V$, the cut-rank function is

$$\rho_G(X)=\operatorname{rank}_{\mathbb F_2}(A_{X,\,V\setminus X}),$$

where $A_{X,\,V\setminus X}$ is the $X\times (V\setminus X)$ submatrix of the adjacency matrix over $\mathbb F_2$. The function $\rho_G$ is a connectivity function: it is symmetric and submodular (DeVos et al., 2019).

Branch-depth is defined for an arbitrary connectivity function $K_tK_t$0 on a finite ground set $K_tK_t$1. A decomposition is a pair $K_tK_t$2 consisting of a tree $K_tK_t$3 with at least one internal node and a bijection $K_tK_t$4 leaves$K_tK_t$5. For an internal node $K_tK_t$6 of $K_tK_t$7, let $K_tK_t$8 be the partition of $K_tK_t$9 induced by the connected components of $t$0. The width of $t$1 is

$t$2

and the width of the decomposition is the maximum over internal nodes. Its radius is the radius of the tree $t$3. A $t$4-decomposition is one of width at most $t$5 and radius at most $t$6. The branch-depth of $t$7 is the minimum $t$8 such that there exists a $t$9-decomposition (DeVos et al., 2019).

Rank-depth is obtained by specializing $G=(V,E)$0 to $G=(V,E)$1: $G=(V,E)$2 Equivalently, rank-depth is the minimum integer $G=(V,E)$3 such that $G=(V,E)$4 has a $G=(V,E)$5-decomposition. If $G=(V,E)$6, the rank-depth is $G=(V,E)$7 (DeVos et al., 2019, Ahn et al., 17 Jul 2025).

This formulation makes the parameter simultaneously “depth-like” and “rank-like.” The tree radius is the depth constraint, while the width term records the maximum binary cut-rank realized by unions of parts around an internal node. In contrast to sparse depth parameters, the complexity across a split is measured by linear algebra rather than incidence.

2. Relation to shrub-depth and tree-depth

The central classification theorem for the parameter is that a class of simple graphs has bounded rank-depth if and only if it has bounded shrub-depth. This is Theorem 4.11 in the branch-depth framework, and it identifies rank-depth as the decomposition parameter that exactly captures the bounded shrub-depth classes (DeVos et al., 2019).

The proof proceeds in both directions. If a graph $G=(V,E)$8 has a $G=(V,E)$9-shrubbery, then its cut-rank function has a $X\subseteq V$0-decomposition. Conversely, for each $X\subseteq V$1, there exists

$X\subseteq V$2

such that if the cut-rank function of a simple graph $X\subseteq V$3 has a $X\subseteq V$4-decomposition, then $X\subseteq V$5 has an $X\subseteq V$6-shrubbery (DeVos et al., 2019).

Rank-depth is analogous to tree-depth, but the clean equivalence is with shrub-depth rather than tree-depth. The paper proves the upper bound

$X\subseteq V$7

when $X\subseteq V$8 has tree-depth $X\subseteq V$9, where $$\rho_G(X)=\operatorname{rank}_{\mathbb F_2}(A_{X,\,V\setminus X}),$$0 is the incidence graph of $$\rho_G(X)=\operatorname{rank}_{\mathbb F_2}(A_{X,\,V\setminus X}),$$1. Thus tree-depth controls rank-depth, but rank-depth can be much smaller (DeVos et al., 2019).

The path family exhibits this separation sharply. For the path $$\rho_G(X)=\operatorname{rank}_{\mathbb F_2}(A_{X,\,V\setminus X}),$$2,

$$\rho_G(X)=\operatorname{rank}_{\mathbb F_2}(A_{X,\,V\setminus X}),$$3

and

$$\rho_G(X)=\operatorname{rank}_{\mathbb F_2}(A_{X,\,V\setminus X}),$$4

Hence rank-depth on paths grows like $$\rho_G(X)=\operatorname{rank}_{\mathbb F_2}(A_{X,\,V\setminus X}),$$5, much more slowly than tree-depth, which is $$\rho_G(X)=\operatorname{rank}_{\mathbb F_2}(A_{X,\,V\setminus X}),$$6 for paths (DeVos et al., 2019).

A common misconception is that rank-depth is simply rank-width with a radius restriction. The theory instead treats it as the branch-depth of the cut-rank connectivity function, with shrub-depth as the corresponding class-level notion. The same source explicitly emphasizes that bounded rank-depth is not especially meaningful for a single graph in the same way as tree-depth is, but captures the structure of whole classes (DeVos et al., 2019).

3. Vertex-minors and the path obstruction theorem

Because local complementation preserves cut-rank, rank-depth is invariant under local equivalence and monotone under vertex-minors: $$\rho_G(X)=\operatorname{rank}_{\mathbb F_2}(A_{X,\,V\setminus X}),$$7 This monotonicity is the reason obstruction theory for rank-depth is naturally phrased in terms of vertex-minors (Kwon et al., 2019).

The decisive obstruction theorem states that for every positive integer $$\rho_G(X)=\operatorname{rank}_{\mathbb F_2}(A_{X,\,V\setminus X}),$$8, there exists an integer $$\rho_G(X)=\operatorname{rank}_{\mathbb F_2}(A_{X,\,V\setminus X}),$$9 such that every graph of rank-depth at least $A_{X,\,V\setminus X}$0 contains a vertex-minor isomorphic to the path $A_{X,\,V\setminus X}$1. Equivalently, large rank-depth is characterized by long paths as vertex-minors (Kwon et al., 2019).

This yields an immediate corollary: for every integer $A_{X,\,V\setminus X}$2, the class of graphs with no vertex-minor isomorphic to $A_{X,\,V\setminus X}$3 has bounded shrub-depth. Combined with the equivalence between bounded rank-depth and bounded shrub-depth, this provides the exact obstruction family for bounded shrub-depth under vertex-minors (Kwon et al., 2019).

The proof is not a single reduction. In the bounded-rank-width regime, the argument builds path-like structures by induction, using lemmas that control the survival of rank-depth under deletions and that glue decompositions across low cut-rank bipartitions. Ramsey theory is then used to regularize the adjacency patterns between the constructed pieces. In the high-rank-width regime, the theorem of Geelen, Kwon, McCarty, and Wollan supplies the desired path vertex-minor because every path is a circle graph (Kwon et al., 2019).

The same paper proves a stronger bipartite statement: for every $A_{X,\,V\setminus X}$4, there exists $A_{X,\,V\setminus X}$5 such that every bipartite graph of rank-depth at least $A_{X,\,V\setminus X}$6 contains a pivot-minor isomorphic to $A_{X,\,V\setminus X}$7. This is a genuine strengthening in the bipartite case, but it does not extend to arbitrary graphs without modification (Kwon et al., 2019).

4. Pivot-minor obstructions and the half-graph clique configuration

A pivot-minor is obtained by a sequence of pivot operations and vertex deletions. For an edge $A_{X,\,V\setminus X}$8, the pivot is

$A_{X,\,V\setminus X}$9

equivalently $X\times (V\setminus X)$0, where local complementation at a vertex toggles adjacency among its neighbors (Ahn et al., 17 Jul 2025).

The pivot-minor obstruction theory for rank-depth is sharper than the earlier bipartite result but also more complicated. For every $X\times (V\setminus X)$1, there exists a function $X\times (V\setminus X)$2 such that every graph of rank-depth at least $X\times (V\setminus X)$3 has a pivot-minor isomorphic to $X\times (V\setminus X)$4 or $X\times (V\setminus X)$5. Here $X\times (V\setminus X)$6 is the graph formed from two disjoint cliques

$X\times (V\setminus X)$7

by adding edges $X\times (V\setminus X)$8 if and only if $X\times (V\setminus X)$9; the bipartite pattern between the cliques is the half graph (Ahn et al., 17 Jul 2025).

This resolves the open problem of Kwon, McCarty, Oum, and Wollan asking whether every pivot-minor-closed graph class of bounded rank-depth is characterized by forbidding, for some $\mathbb F_2$0, both $\mathbb F_2$1 and $\mathbb F_2$2. The theorem gives exactly that characterization (Ahn et al., 17 Jul 2025).

The proof uses a theorem of Mählmann: there exists a function $\mathbb F_2$3 such that every graph of rank-depth at least $\mathbb F_2$4 has an induced subgraph isomorphic to one of $\mathbb F_2$5, $\mathbb F_2$6, $\mathbb F_2$7, or a flipped $\mathbb F_2$8. Each case is then converted to a pivot-minor $\mathbb F_2$9 or $\rho_G$0 (Ahn et al., 17 Jul 2025).

This pivot-minor theorem also clarifies a limitation of path-only formulations. The class

$\rho_G$1

has unbounded rank-depth, but none of these graphs has a pivot-minor isomorphic to $\rho_G$2. A plausible implication is that, outside the bipartite setting, long paths alone do not control rank-depth under pivot-minors; the dense half-graph clique configuration $\rho_G$3 is essential (Kwon et al., 2019).

5. Depth-$\rho_G$4 rank-brittleness as a refinement

Depth-$\rho_G$5 rank-brittleness is a refinement of rank-depth obtained by fixing the decomposition radius at $\rho_G$6 and minimizing a different width parameter. A decomposition is a pair $\rho_G$7 where $\rho_G$8 is a tree, $\rho_G$9 is a bijection from $K_tK_t$00 to the leaves of $K_tK_t$01, and $K_tK_t$02 has radius at most $K_tK_t$03. If an internal node $K_tK_t$04 of $K_tK_t$05 induces parts $K_tK_t$06 of $K_tK_t$07, then the width of $K_tK_t$08 is

$K_tK_t$09

The minimum width over all such decompositions of radius at most $K_tK_t$10 is the depth-$K_tK_t$11 rank-brittleness, denoted $K_tK_t$12 in the paper. In particular,

$K_tK_t$13

and

$K_tK_t$14

Rank-depth is controlled by these parameters in the sense that rank-depth is at most $K_tK_t$15 for $K_tK_t$16 (Kwon et al., 2019).

The radius-$K_tK_t$17 case admits a precise forbidden-vertex-minor characterization. For a vertex-minor ideal $K_tK_t$18, the class has bounded depth-$K_tK_t$19 rank-brittleness if and only if

$K_tK_t$20

where $K_tK_t$21 denotes the $K_tK_t$22-subdivision of $K_tK_t$23 and $K_tK_t$24 is the disjoint union of $K_tK_t$25 copies of it. Equivalently, bounded depth-$K_tK_t$26 rank-brittleness is characterized by excluding both all paths and all disjoint unions $K_tK_t$27 (Kwon et al., 2019).

The engine is an obstruction theorem: for every fixed $K_tK_t$28, there exists a threshold $K_tK_t$29 such that every graph with depth-$K_tK_t$30 rank-brittleness at least $K_tK_t$31 contains, as a vertex-minor, either $K_tK_t$32 or $K_tK_t$33. The proof first forces large structured vertex-minors of the form $K_tK_t$34, and then uses Ramsey-type arguments, sunflower lemmas, and local complementations to reduce them to paths or subdivided stars (Kwon et al., 2019).

This refinement also sits inside the broader width hierarchy. The paper proves

$K_tK_t$35

Therefore bounded depth-$K_tK_t$36 rank-brittleness implies bounded rank-depth, and bounded rank-depth implies bounded linear rank-width. As a corollary, for every fixed $K_tK_t$37, graphs with no vertex-minor isomorphic to $K_tK_t$38 have bounded depth-$K_tK_t$39 rank-brittleness, bounded rank-depth, and bounded linear rank-width (Kwon et al., 2019).

6. Structural consequences and matroid connections

The equivalence with shrub-depth has several immediate structural consequences. Since bounded shrub-depth classes are well-quasi-ordered by induced subgraphs, bounded rank-depth classes are also well-quasi-ordered by induced subgraphs. In particular, for every $K_tK_t$40, there is a finite list of graphs such that a graph $K_tK_t$41 has rank-depth at most $K_tK_t$42 if and only if no graph in the list is isomorphic to an induced subgraph of $K_tK_t$43 (DeVos et al., 2019).

Rank-depth also has a direct binary matroid interpretation. If $K_tK_t$44 is a fundamental graph of a binary matroid $K_tK_t$45, then the cut-rank function of $K_tK_t$46 matches the connectivity function of $K_tK_t$47. Consequently, if $K_tK_t$48 is a fundamental graph of a binary matroid $K_tK_t$49, then the branch-depth of $K_tK_t$50 is equal to the rank-depth of $K_tK_t$51. This is the graph–matroid bridge underlying later minor consequences (DeVos et al., 2019).

Using the pivot-minor path theorem in the bipartite setting, one obtains a corollary for binary matroids: for every $K_tK_t$52, there exists $K_tK_t$53 such that every binary matroid of branch-depth at least $K_tK_t$54 contains a minor isomorphic to $K_tK_t$55, where $K_tK_t$56 is the fan graph obtained by adding a universal vertex to $K_tK_t$57, and $K_tK_t$58 is its cycle matroid (Kwon et al., 2019).

Taken together, these results place rank-depth in a dense analogue of the classical tree-depth picture. Tree-depth is controlled by subgraph paths; rank-depth is controlled by vertex-minor paths; and under pivot-minors the unavoidable structures are paths together with the half-graph clique configuration $K_tK_t$59. The depth-$K_tK_t$60 theory shows that once the radius is fixed, additional obstructions such as $K_tK_t$61 emerge. This suggests a layered hierarchy in which the cut-rank decomposition formalism, rather than a single forbidden configuration, is the organizing principle.