---
title: Rank-depth in Graph Theory
url: https://www.emergentmind.com/topics/rank-depth
type: topic
---

# Rank-depth in Graph Theory

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 [1903.11988] [1911.00230] [2507.12697].

## 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 [1903.11988].

Branch-depth is defined for an arbitrary connectivity function \(\lambda\) on a finite ground set \(E\). A decomposition is a pair \((T,\sigma)\) consisting of a tree \(T\) with at least one internal node and a bijection \(\sigma:E\to\) leaves\((T)\). For an internal node \(v\) of \(T\), let \(P_v\) be the partition of \(E\) induced by the connected components of \(T-v\). The width of \(v\) is
\[
\lambda(P_v)=\max_{Q\subseteq P_v}\lambda\!\left(\bigcup Q\right),
\]
and the width of the decomposition is the maximum over internal nodes. Its radius is the radius of the tree \(T\). A \((k,r)\)-decomposition is one of width at most \(k\) and radius at most \(r\). The branch-depth of \(\lambda\) is the minimum \(k\) such that there exists a \((k,k)\)-decomposition [1903.11988].

Rank-depth is obtained by specializing \(\lambda\) to \(\rho_G\):
\[
\operatorname{rd}(G):=\text{branch-depth of }\rho_G.
\]
Equivalently, rank-depth is the minimum integer \(k\) such that \(G\) has a \((k,k)\)-decomposition. If \(|V(G)|\le 1\), the rank-depth is \(0\) [1903.11988] [2507.12697].

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 [1903.11988].

The proof proceeds in both directions. If a graph \(G\) has a \((k,d)\)-shrubbery, then its cut-rank function has a \((k,d)\)-decomposition. Conversely, for each \(k,r\), there exists
\[
a=(1+o(1))\,2^{\left(22^{k+1}(2^{2k+2}-1)+1\right)r/2}
\]
such that if the cut-rank function of a simple graph \(G\) has a \((k,r)\)-decomposition, then \(G\) has an \((a,r)\)-shrubbery [1903.11988].

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
\[
\operatorname{rd}(G)\le \operatorname{rd}(I(G))\le t
\]
when \(G\) has tree-depth \(t\), where \(I(G)\) is the incidence graph of \(G\). Thus tree-depth controls rank-depth, but rank-depth can be much smaller [1903.11988].

The path family exhibits this separation sharply. For the path \(P_n\),
\[
\operatorname{rd}(P_n)>\frac{\log n}{\log(1+4\log n)}
\]
and
\[
\operatorname{rd}(P_n)\le (1+o(1))\frac{\log n}{\log\log n}.
\]
Hence rank-depth on paths grows like \(\log n/\log\log n\), much more slowly than tree-depth, which is \(\Theta(\log n)\) for paths [1903.11988].

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 [1903.11988].

## 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:
\[
H\text{ a vertex-minor of }G \Longrightarrow \operatorname{rd}(H)\le \operatorname{rd}(G).
\]
This monotonicity is the reason obstruction theory for rank-depth is naturally phrased in terms of vertex-minors [1911.00230].

The decisive obstruction theorem states that 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 \(P_t\). Equivalently, large rank-depth is characterized by long paths as vertex-minors [1911.00230].

This yields an immediate corollary: for every integer \(t\), the class of graphs with no vertex-minor isomorphic to \(P_t\) 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 [1911.00230].

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 [1911.00230].

The same paper proves a stronger bipartite statement: for every \(t\), there exists \(N(t)\) such that every bipartite graph of rank-depth at least \(N(t)\) contains a pivot-minor isomorphic to \(P_t\). This is a genuine strengthening in the bipartite case, but it does not extend to arbitrary graphs without modification [1911.00230].

## 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 \(uv\), the pivot is
\[
G\wedge uv:=G*u*v*u,
\]
equivalently \(G*v*u*v\), where local complementation at a vertex toggles adjacency among its neighbors [2507.12697].

The pivot-minor obstruction theory for rank-depth is sharper than the earlier bipartite result but also more complicated. For every \(t\in\mathbb N\), there exists a function \(f:\mathbb N\to\mathbb N\) such that every graph of rank-depth at least \(f(t)\) has a pivot-minor isomorphic to \(P_t\) or \(K_tK_t\). Here \(K_tK_t\) is the graph formed from two disjoint cliques
\[
u_1,\dots,u_t \qquad\text{and}\qquad v_1,\dots,v_t
\]
by adding edges \(u_iv_j\) if and only if \(i\ge j\); the bipartite pattern between the cliques is the half graph [2507.12697].

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 \(t\), both \(P_t\) and \(K_tK_t\). The theorem gives exactly that characterization [2507.12697].

The proof uses a theorem of Mählmann: there exists a function \(g:\mathbb N\to\mathbb N\) such that every graph of rank-depth at least \(g(s)\) has an induced subgraph isomorphic to one of \(K_sK_s\), \(K_s\overline{K_s}\), \(\overline{K_s}\overline{K_s}\), or a flipped \(sP_s\). Each case is then converted to a pivot-minor \(P_t\) or \(K_tK_t\) [2507.12697].

This pivot-minor theorem also clarifies a limitation of path-only formulations. The class
\[
\{K_n \, Z \, K_n : n\ge 1\}
\]
has unbounded rank-depth, but none of these graphs has a pivot-minor isomorphic to \(P_5\). 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 \(K_tK_t\) is essential [1911.00230].

## 5. Depth-\(2\) rank-brittleness as a refinement

Depth-\(2\) rank-brittleness is a refinement of rank-depth obtained by fixing the decomposition radius at \(2\) and minimizing a different width parameter. A decomposition is a pair \((T,o)\) where \(T\) is a tree, \(o\) is a bijection from \(V(G)\) to the leaves of \(T\), and \(T\) has radius at most \(d\). If an internal node \(v\) of \(T\) induces parts \(X_1,\dots,X_m\) of \(V(G)\), then the width of \(v\) is
\[
\max_i \rho_G(X_i).
\]
The minimum width over all such decompositions of radius at most \(d\) is the depth-\(d\) rank-brittleness, denoted \(\operatorname{rbrit}_d(G)\) in the paper. In particular,
\[
\operatorname{rbrit}_1(G)=\max_{A\subseteq V(G)}\rho_G(A)
\]
and
\[
\operatorname{rbrit}_1(G) > \operatorname{rbrit}_2(G) > \operatorname{rbrit}_3(G) > \cdots.
\]
Rank-depth is controlled by these parameters in the sense that rank-depth is at most \(\max\{d,\operatorname{rbrit}_d(G)\}\) for \(d\ge 1\) [1906.05753].

The radius-\(2\) case admits a precise forbidden-vertex-minor characterization. For a vertex-minor ideal \(\mathcal C\), the class has bounded depth-\(2\) rank-brittleness if and only if
\[
\{P_1,P_2,P_3,\dots\}\not\subseteq \mathcal C
\quad\text{and}\quad
\{T_{2,1},\,2T_{2,2},\,3T_{2,3},\,4T_{2,4},\dots\}\not\subseteq \mathcal C,
\]
where \(T_{2,n}\) denotes the \(1\)-subdivision of \(K_{1,n}\) and \(nT_{2,n}\) is the disjoint union of \(n\) copies of it. Equivalently, bounded depth-\(2\) rank-brittleness is characterized by excluding both all paths and all disjoint unions \(nT_{2,n}\) [1906.05753].

The engine is an obstruction theorem: for every fixed \(n\), there exists a threshold \(N\) such that every graph with depth-\(2\) rank-brittleness at least \(N\) contains, as a vertex-minor, either \(P_n\) or \(nT_{2,n}\). The proof first forces large structured vertex-minors of the form \(nK_k\), and then uses Ramsey-type arguments, sunflower lemmas, and local complementations to reduce them to paths or subdivided stars [1906.05753].

This refinement also sits inside the broader width hierarchy. The paper proves
\[
\operatorname{lrw}(G)\le \operatorname{rd}(G)^2.
\]
Therefore bounded depth-\(2\) rank-brittleness implies bounded rank-depth, and bounded rank-depth implies bounded linear rank-width. As a corollary, for every fixed \(n\), graphs with no vertex-minor isomorphic to \(nP_5\) have bounded depth-\(2\) rank-brittleness, bounded rank-depth, and bounded linear rank-width [1906.05753].

## 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 \(k\), there is a finite list of graphs such that a graph \(G\) has rank-depth at most \(k\) if and only if no graph in the list is isomorphic to an induced subgraph of \(G\) [1903.11988].

Rank-depth also has a direct binary matroid interpretation. If \(G\) is a fundamental graph of a binary matroid \(M\), then the cut-rank function of \(G\) matches the connectivity function of \(M\). Consequently, if \(G\) is a fundamental graph of a binary matroid \(M\), then the branch-depth of \(M\) is equal to the rank-depth of \(G\). This is the graph–matroid bridge underlying later minor consequences [1903.11988].

Using the pivot-minor path theorem in the bipartite setting, one obtains a corollary for binary matroids: for every \(t\), there exists \(N(t)\) such that every binary matroid of branch-depth at least \(N(t)\) contains a minor isomorphic to \(M(F_t)\), where \(F_t\) is the fan graph obtained by adding a universal vertex to \(P_t\), and \(M(F_t)\) is its cycle matroid [1911.00230].

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 \(K_tK_t\). The depth-\(2\) theory shows that once the radius is fixed, additional obstructions such as \(nT_{2,n}\) emerge. This suggests a layered hierarchy in which the cut-rank decomposition formalism, rather than a single forbidden configuration, is the organizing principle.

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