Treedepth: Graph Hierarchy & Applications

• Treedepth is a graph parameter defined as the minimum height of a rooted forest whose closure contains every graph edge, linking structural properties with algorithmic insights.
• Its equivalent formulations—as elimination tree height, vertex ranking, and centered coloring—provide diverse methods for analyzing and decomposing graphs.
• Treedepth is central to fixed-parameter tractability and obstruction theory, influencing the design of exact algorithms and complexity bounds for NP-hard problems.

Searching arXiv for recent and foundational treedepth papers to ground the article. Treedepth is a graph parameter that measures the minimum hierarchical height needed to represent a graph by a rooted forest whose ancestor–descendant closure contains every edge of the graph. In the standard formulation, a treedepth decomposition of a graph $G$ is a rooted forest $F$ on $V(G)$ such that every edge $uv \in E(G)$ has one endpoint as an ancestor of the other in $F$; the treedepth $\td(G)$ is the minimum height of such a forest, where height is the maximum number of vertices on a root–leaf path (Trimble, 2020). Equivalently, for a rooted forest $F$, the closure $\operatorname{clos}(F)$ joins every vertex to its ancestors, and $\td(G)$ is the minimum height of $F$ with $F$0 (Hodor et al., 6 Aug 2025). Treedepth is also known under equivalent formulations such as elimination tree height, vertex ranking, and centered coloring (Chen et al., 2016). It occupies a central position between sparse graph structure and exact or fixed-parameter algorithmics, while also admitting generalizations such as $F$1-treedepth and $F$2-treedepth that interpolate toward treewidth (Hodor et al., 6 Aug 2025, Rambaud, 20 Jan 2025).

1. Definitions and equivalent formulations

For a graph $F$3, the recursive definition used in recent structural work is (Hodor et al., 6 Aug 2025)

$F$4

This matches the elimination-tree viewpoint: for connected $F$5, one chooses a root vertex $F$6, recursively decomposes each component of $F$7, and attaches their roots below $F$8 (Trimble, 2020).

The rooted-forest formulation is standard. A rooted forest $F$9 on $V(G)$0 has closure $V(G)$1, where two vertices are adjacent if one is an ancestor of the other. Then (Hodor et al., 6 Aug 2025)

$V(G)$2

The same concept appears in algorithmic papers as an elimination forest: a rooted forest $V(G)$3 such that for every edge $V(G)$4, either $V(G)$5 is an ancestor of $V(G)$6 or $V(G)$7 is an ancestor of $V(G)$8 (Chen et al., 2016, Trimble, 2020).

Several equivalent characterizations are explicitly noted. Treedepth equals the minimum number of colors in a vertex ranking and in a centered coloring (Chen et al., 2016, Kun et al., 2018). In the centered-coloring formulation, every connected subgraph must contain a uniquely colored vertex; in the vertex-ranking formulation, every path whose endpoints have the same color must contain an interior vertex with a higher color (Kun et al., 2018). These equivalences connect treedepth simultaneously to graph minors, sparse graph classes, and decomposition-based algorithm design.

Basic properties recorded in the literature include the following. One has $V(G)$9 if and only if $uv \in E(G)$0 has no edges, and treedepth is monotone under taking subgraphs: if $uv \in E(G)$1, then $uv \in E(G)$2 (Hodor et al., 6 Aug 2025). For the path $uv \in E(G)$3,

$uv \in E(G)$4

as a standard fact used throughout the area (Hodor et al., 6 Aug 2025). A folklore characterization states that a minor-closed class has bounded treedepth if and only if it excludes some path as a minor, equivalently as a subgraph (Hodor et al., 6 Aug 2025). This criterion is generalized in the $uv \in E(G)$5-treedepth framework discussed below (Rambaud, 20 Jan 2025).

2. Structural position relative to treewidth, pathwidth, and $uv \in E(G)$6-treedepth

Treedepth is more restrictive than treewidth and pathwidth. A standard inequality used in algorithmic work is (Chen et al., 2016)

$uv \in E(G)$7

A related formulation appearing in more recent work is (Bergougnoux et al., 22 Oct 2025)

$uv \in E(G)$8

up to indexing conventions. These inequalities encode the fact that bounded treedepth implies bounded pathwidth and bounded treewidth, but not conversely; long paths provide the canonical separation, since they have treewidth $uv \in E(G)$9 and unbounded treedepth (Hodor et al., 6 Aug 2025, Czerwiński et al., 2019).

Treedepth has a natural recursive generalization. For every positive integer $F$0, the parameter $F$1-treedepth $F$2 is defined as the largest graph parameter satisfying (Rambaud, 20 Jan 2025):

1. $F$3,
2. $F$4 for every graph $F$5 and vertex $F$6,
3. $F$7 whenever $F$8 is a $F$9-clique-sum of $\td(G)$0 and $\td(G)$1.

This produces a monotone hierarchy

$\td(G)$2

(Rambaud, 20 Jan 2025). The case $\td(G)$3 is the $\td(G)$4-treedepth of Huynh–Joret–Micek–Seweryn–Wollan, later central to induced-path questions (Hodor et al., 6 Aug 2025). The case $\td(G)$5 coincides with treewidth plus one (Rambaud, 20 Jan 2025). This suggests that treedepth may be viewed as the most restrictive member of a clique-sum-compatible width/depth hierarchy.

A decomposition-based characterization also exists for $\td(G)$6-treedepth. A graph $\td(G)$7 has $\td(G)$8-treedepth at most $\td(G)$9 if and only if $F$0 admits a $F$1-dismantable tree decomposition of width $F$2, equivalently

$F$3

(Rambaud, 20 Jan 2025). This places ordinary treedepth and treewidth in a single axiomatic framework.

3. Variants, refinements, and obstruction viewpoints

A prominent refinement is $F$4-treedepth, denoted $F$5, defined recursively on blocks rather than connected components (Hodor et al., 6 Aug 2025): $F$6 Always,

$F$7

(Hodor et al., 6 Aug 2025). The gap can be large: for the path $F$8, treedepth is unbounded with $F$9, but $\operatorname{clos}(F)$0 because every block is an edge or isolated vertex (Hodor et al., 6 Aug 2025). In contrast, on highly connected graphs such as complete graphs or nontrivial complete bipartite graphs, $\operatorname{clos}(F)$1 and $\operatorname{clos}(F)$2 coincide because the graph consists of a single block (Hodor et al., 6 Aug 2025).

The obstruction theory of treedepth is governed by minor-monotonicity. The class $\operatorname{clos}(F)$3 is minor-closed, hence by the Graph Minor Theorem it has a finite set of forbidden minors (Kühn, 1 Dec 2025). A conjecture of Dvořák, Giannopoulou, and Thilikos states that every forbidden minor for treedepth at most $\operatorname{clos}(F)$4 has at most $\operatorname{clos}(F)$5 vertices; analogous claims are stated for forbidden subgraphs and forbidden induced subgraphs (Kühn, 1 Dec 2025). An algorithm computing obstruction sets up to a prescribed size was developed in 2025. Applied to $\operatorname{clos}(F)$6 and $\operatorname{clos}(F)$7, it enumerates 1546 forbidden minors, 1718 forbidden subgraphs, and 12204 forbidden induced subgraphs (Kühn, 1 Dec 2025). Assuming the conjectured $\operatorname{clos}(F)$8 bound, these are the complete obstruction sets for graphs of treedepth at most $\operatorname{clos}(F)$9 (Kühn, 1 Dec 2025).

The excluded-minor approximation viewpoint gives another structural lens. If treedepth is large, then either treewidth is large or the graph contains a tree-like witness of large treedepth. In particular, there exists a constant $\td(G)$0 such that if $\td(G)$1, then either $\td(G)$2 or $\td(G)$3 contains a subcubic tree of treedepth at least $\td(G)$4 as a subgraph (Czerwiński et al., 2019). As a corollary, every graph of treedepth $\td(G)$5 either has treewidth at least $\td(G)$6, contains a subdivision of a full binary tree of depth $\td(G)$7, or contains a path of length $\td(G)$8 (Czerwiński et al., 2019). This improves an earlier $\td(G)$9 bound of Kawarabayashi and Rossman (Czerwiński et al., 2019).

4. Treedepth, induced paths, and hereditary structure

One of the most developed recent directions concerns the interaction of treedepth with hereditary classes defined by forbidden induced paths. The motivating observation is that $F$0 ignores long tree-like chains that treedepth detects, so one may ask when treedepth is controlled by $F$1 on a hereditary class (Hodor et al., 6 Aug 2025).

The main result in this direction is that for $F$2-free graphs, treedepth is bounded by an explicit function of $F$3-treedepth. If $F$4, $F$5, and $F$6 is $F$7-free with $F$8, then (Hodor et al., 6 Aug 2025)

$F$9

A sharper technical bound is first proved for $F$00: $F$01 (Hodor et al., 6 Aug 2025). This is nontrivial because on arbitrary hereditary classes containing all paths, no function $F$02 can satisfy $F$03: paths have bounded $F$04 but unbounded $F$05 (Hodor et al., 6 Aug 2025).

The dependence on $F$06 is essentially optimal. For every fixed $F$07, there exists $F$08 such that for every $F$09 there is a $F$10-free graph $F$11 with $F$12 and

$F$13

(Hodor et al., 6 Aug 2025). A more explicit construction yields $F$14-free graphs $F$15 with $F$16 and $F$17 (Hodor et al., 6 Aug 2025). The recursive construction is based on attaching many copies of a lower-level graph $F$18 to the vertices of a higher-level graph $F$19, with the bound

$F$20

for graphs of the specified attachment type (Hodor et al., 6 Aug 2025).

Very short forbidden paths lead to qualitatively stronger relations. If $F$21 is $F$22-free and $F$23, then $F$24. If $F$25 is $F$26-free and $F$27, then

$F$28

and this quadratic bound is tight (Hodor et al., 6 Aug 2025). This suggests a broader conjecture formulated there: for every fixed $F$29, the function

$F$30

may be polynomial in $F$31, although this is only proved for $F$32 (Hodor et al., 6 Aug 2025).

The same work also studies forcing long induced paths from long ordinary paths. For the class $F$33 of graphs of pathwidth at most $F$34, if $F$35 is the largest integer such that every graph in the class containing a path $F$36 as a subgraph also contains an induced path of order at least $F$37, then

$F$38

(Hodor et al., 6 Aug 2025). For the class $F$39,

$F$40

(Hodor et al., 6 Aug 2025). These results connect treedepth-like parameters to induced-subgraph forcing phenomena.

5. Algorithmic roles: exact computation, space complexity, and meta-theorems

Treedepth is algorithmically important both as a parameter and as an object to compute. Exact computation remains challenging. A practical exact solver, Bute-Plus, is based on a positive-instance driven dynamic program that builds minimum-depth elimination trees bottom-up (Trimble, 2020). For the decision problem “does there exist an elimination tree of depth $F$41?”, it computes families $F$42 of connected vertex sets whose induced subgraphs have treedepth at most $F$43, whose neighborhood has size $F$44, and which can appear as subtrees rooted at depth $F$45 in some elimination tree of depth $F$46 (Trimble, 2020). The solver is strengthened by a specialized trie data structure, a domination rule, and a heuristic presolve step (Trimble, 2020).

The computational complexity landscape sharpened considerably in 2025. Exactly computing treedepth is NP-complete, and earlier reductions only excluded $F$47-time exact algorithms under ETH. A direct reduction from Satisfiability now shows that 1.0003-approximating treedepth is NP-hard, and that exactly computing treedepth on an $F$48-vertex graph requires time $F$49 unless ETH fails (Bonnet et al., 18 Jul 2025). The same work derives constants $F$50 such that any $F$51-approximation requires time $F$52 under ETH (Bonnet et al., 18 Jul 2025). This rules out a PTAS unless $F$53 (Bonnet et al., 18 Jul 2025).

As a parameter for algorithms on hard graph problems, treedepth supports different time–space tradeoffs than treewidth. Dynamic programming on treedepth decompositions inherits strong space lower bounds: any single-pass DP on decompositions of width or depth $F$54 cannot solve Dominating Set or 3-Coloring using $F$55 space, nor Vertex Cover using $F$56 space, for any $F$57 (Chen et al., 2016). These lower bounds are proved via Myhill–Nerode families of bounded treedepth and also imply analogous space lower bounds for treewidth and pathwidth (Chen et al., 2016). At the same time, treedepth enables branching or hybrid branching-DP algorithms with low space usage: for example, Dominating Set can be solved in time $F$58 using $F$59 space, or in time $F$60 using $F$61 space, where $F$62 is the depth of the treedepth decomposition (Chen et al., 2016).

A major recent development is a logic-based meta-theorem. A tailored logic, $F$63, extends fully existential $F$64 with neighborhood operators $F$65, acyclicity, connectivity, and clique predicates, and captures a broad class of NP-hard problems known to admit single-exponential FPT algorithms on treedepth (Bergougnoux et al., 22 Oct 2025). Given an elimination tree of depth $F$66, model checking for $F$67 runs in time

$F$68

and space

$F$69

where $F$70 and $F$71 are parameters of the finitely recognizable neighborhood sets used in the formula (Bergougnoux et al., 22 Oct 2025). For full $F$72, there is a Monte Carlo algorithm with the same single-exponential dependence and polynomial space (Bergougnoux et al., 22 Oct 2025). The logic captures Independent Set, Hamiltonian Cycle, various $F$73-domination problems, CNF-SAT via incidence graphs, and others (Bergougnoux et al., 22 Oct 2025).

Connectivity problems parameterized by treedepth also admit specialized improvements. A 2026 algorithm solves Hamiltonian Cycle, Hamiltonian Path, Long Cycle, Long Path, Minimum Cycle Cover, and Partial Cycle Cover in time $F$74 and polynomial space when given an elimination forest of depth $F$75, improving a previous $F$76 bound (Kratsch, 25 Mar 2026). The core technical device is counting ordered pairs of consistent matchings via inclusion–exclusion structured by the elimination forest (Kratsch, 25 Mar 2026).

6. Applications beyond pure graph structure

Treedepth appears in integer programming through structural parameters of the constraint matrix. For a matrix $F$77, one may consider the primal, dual, and incidence Gaifman graphs $F$78, $F$79, and $F$80, with corresponding treedepth parameters $F$81, $F$82, and $F$83 (Eiben et al., 2020). Earlier results showed fixed-parameter tractability of ILP feasibility with respect to primal or dual treedepth together with the largest absolute coefficient of $F$84 (Eiben et al., 2020). However, this does not extend to incidence treedepth: deciding feasibility of

$F$85

is NP-hard even when $F$86, $F$87, and $F$88 (Eiben et al., 2020). This demonstrates that incidence treedepth is too weak a restriction to recover the same tractability phenomenon.

The concept also extends naturally to weighted variants. A weighted treedepth decomposition assigns weights $F$89 and measures the maximum total weight on a root–leaf path rather than the number of vertices. Weighted treedepth is NP-complete even on bounded-degree graphs: specifically, on graphs of maximum degree $F$90 (Dirks et al., 21 Oct 2025). On the positive side, it is polynomial-time solvable on paths and on 1-subdivided stars (Dirks et al., 21 Oct 2025). This indicates that weighted treedepth is substantially more delicate than the unweighted parameter on sparse classes.

Coloring-based applications form another line of work. Low-treedepth colorings, especially $F$91-centered colorings, are fundamental in bounded-expansion graph classes because any connected subgraph using fewer than $F$92 colors has bounded treedepth (Kun et al., 2018). A weaker notion, $F$93-linear coloring, only requires this condition on paths. For linear colorings, the induced treedepth can be larger, but there are polynomial bounds. In general graphs, treedepth is polynomially bounded in the linear-coloring number (Kun et al., 2018). For trees, Schaffer’s vertex-ranking algorithm yields

$F$94

where $F$95 is the maximum degree, and the factor $F$96 is asymptotically tight for binary trees (Kun et al., 2018). For interval graphs, a linear coloring with $F$97 colors can be converted in polynomial time into a centered coloring with at most $F$98 colors (Kun et al., 2018). These results make explicit how treedepth mediates between path-based and connected-subgraph-based colorings.

7. Random graphs and probabilistic behavior

The asymptotic behavior of treedepth on random graphs has been determined in the classical Erdős–Rényi model. If $F$99 with $V(G)$00, then asymptotically almost surely

$V(G)$01

(Perarnau et al., 2011). Thus dense random graphs have treedepth very close to the maximum possible value $V(G)$02, matching the intuition that they behave almost like cliques.

In the sparse regime $V(G)$03, the behavior exhibits a phase transition (Perarnau et al., 2011). If $V(G)$04, then asymptotically almost surely $V(G)$05. If $V(G)$06, then $V(G)$07. If $V(G)$08, then $V(G)$09 (Perarnau et al., 2011). The supercritical linear regime is derived via the existence of a linear-size edge-expander subgraph and the consequent linear lower bound on treewidth (Perarnau et al., 2011). At the critical point $V(G)$10, every width parameter bounded by a function of treewidth is asymptotically almost surely constant, because $V(G)$11, while treedepth remains $V(G)$12 due to the diameter of the largest component (Perarnau et al., 2011). Random $V(G)$13-regular graphs for fixed $V(G)$14 also have linear tree-depth (Perarnau et al., 2011).

This probabilistic picture underscores a recurring theme: treedepth is more sensitive than treewidth to long paths and diameter. At the critical threshold, treewidth collapses to $V(G)$15 while treedepth still grows logarithmically (Perarnau et al., 2011).

8. Conceptual synthesis and open directions

Several themes recur across the literature. Treedepth measures a kind of hierarchical shallowness stronger than treewidth or pathwidth, and this extra rigidity can be exploited algorithmically, especially for low-space exact algorithms (Chen et al., 2016, Kratsch, 25 Mar 2026, Bergougnoux et al., 22 Oct 2025). At the same time, exact computation and tight approximation are hard in a strong sense (Bonnet et al., 18 Jul 2025). Generalizations such as $V(G)$16-treedepth and $V(G)$17-treedepth show that treedepth belongs to a broader family interpolating toward treewidth (Hodor et al., 6 Aug 2025, Rambaud, 20 Jan 2025).

Several open directions are explicitly identified. One longstanding question is whether exact treedepth can be computed in time $V(G)$18 when parameterized by $V(G)$19; the best known exact algorithms remain $V(G)$20 (Bonnet et al., 18 Jul 2025). In the hereditary setting of forbidden induced paths, it is open whether $V(G)$21 is polynomial in $V(G)$22 for every fixed $V(G)$23 (Hodor et al., 6 Aug 2025). In obstruction theory, the Dvořák–Giannopoulou–Thilikos conjecture asserting a $V(G)$24-vertex bound on treedepth obstructions remains unresolved (Kühn, 1 Dec 2025). On the algorithmic meta-theory side, open problems include removing the dependence of the polynomial exponent on size measurements and acyclicity constraints in treedepth meta-theorems, derandomizing the full $V(G)$25 algorithm, and extending the framework to model counting (Bergougnoux et al., 22 Oct 2025).

A plausible implication of these developments is that treedepth functions less as a minor variant of treewidth than as a distinct organizing principle for shallow graph structure. It is simultaneously a decomposition depth, a ranking number, a centered-coloring number, and the first member of a clique-sum-sensitive hierarchy ending at treewidth (Chen et al., 2016, Kun et al., 2018, Rambaud, 20 Jan 2025). Its continued relevance in sparse graph theory, induced-subgraph structure, exact algorithms, and integer programming indicates that it has become one of the standard structural parameters through which graph hierarchy is analyzed.