Weak Shallow Minors in Graph Theory

• Weak shallow minors are conceptually linked to standard shallow minors, without a distinct formal definition in the literature.
• They encompass models based on bounded-depth contraction, topological modifications, and vertex operations, bridging subgraph and full minor regimes.
• These notions underpin key graph algorithms and separator theorems, influencing studies of grid graphs, hereditary classes, and weak colouring numbers.

Searching arXiv for the cited papers and nearby terminology to ground the article. arXiv search: "Weak Shallow Minors (Eppstein, 2013, Wulff-Nilsen, 2014, Wulff-Nilsen, 2011, Hickingbotham et al., 2021, Buffière et al., 2024, Muzi et al., 2017, Paul et al., 2024, Giannopoulou et al., 2011)" Weak shallow minors do not appear in the cited literature as a single standardized formal notion. The nearest recurring concept is the standard shallow minor, also called a minor of bounded depth or a depth-$r$ minor, in which a graph is modeled by pairwise disjoint connected branch sets of bounded radius or bounded diameter. Around that core notion, the literature develops several adjacent themes: quantitative bounded-depth minor containment in grids, separator theorems for graphs excluding bounded-depth clique minors, transfer principles from shallow-minor representations to weak colouring numbers, and distinct hereditary or surface-based variants such as shallow vertex minors and shallow-vortex minors [(Eppstein, 2013); (Wulff-Nilsen, 2014); (Wulff-Nilsen, 2011); (Hickingbotham et al., 2021); (Buffière et al., 2024); (Paul et al., 2024)].

1. Terminological scope

In the cited papers, the phrase “weak shallow minor” is not introduced as a separate formal graph-theoretic definition. Instead, three nearby formal notions recur: standard shallow minors, shallow topological minors, and shallow vertex minors. The first is the closest match when bounded-depth contraction is the operative idea; the third is a different hereditary analogue based on local complementation rather than contraction; and the second lies between subgraphs and shallow minors in the usual containment hierarchy [(Eppstein, 2013); (Hickingbotham et al., 2021); (Buffière et al., 2024); (Muzi et al., 2017)].

Notion	Defining feature in the cited literature	Representative source
Shallow minor / depth-$r$ minor	Disjoint connected branch sets of bounded radius or bounded diameter	(Eppstein, 2013, Wulff-Nilsen, 2014, Wulff-Nilsen, 2011)
Shallow topological minor	A subgraph isomorphic to a bounded subdivision	(Hickingbotham et al., 2021, Muzi et al., 2017)
Shallow vertex minor	$G\ast I_1\ast\dots\ast I_c-D$, with each $I_i$ independent in the current graph	(Buffière et al., 2024)
Shallow-vortex minor	A minor of some shallow-vortex grid $V_k$	(Paul et al., 2024)

The standard shallow-minor definition appears in two slightly different parameterizations. In the damaged-grid setting, a shallow minor at depth $\lambda$ is obtained by contracting disjoint vertex sets $V_i$, each inducing a subgraph of radius at most $\lambda$; depth $0$ gives subgraphs, and sufficiently large depth recovers ordinary minors (Eppstein, 2013). In separator theorems for shallow-minor-free graphs, a graph $H$ is a minor of depth $r$0 if the branch sets have radius at most $r$1 (Wulff-Nilsen, 2014). A closely related formulation uses diameter: a depth-$r$2 minor is a minor model in which each branch set has diameter at most $r$3 (Wulff-Nilsen, 2011). The papers treat these as the operative bounded-depth minor notions; none of them isolates an additional “weak shallow minor” variant.

2. Standard shallow minors as bounded-depth containment

The most direct formal account of shallow minors in the cited corpus is the bounded-depth contraction model. For an $r$4 grid graph with vertex set

$r$5

a depth-$r$6 shallow minor is obtained by contracting pairwise disjoint connected subgraphs of radius at most $r$7, then deleting vertices and possibly deleting resulting edges. In that setting, shallow minors interpolate between subgraphs and unrestricted minors: depth $r$8 is exactly the subgraph case, while sufficiently large depth yields ordinary minor behavior (Eppstein, 2013).

This interpolation is quantified sharply for damaged two-dimensional grids. If $r$9 vertices are deleted from an $G\ast I_1\ast\dots\ast I_c-D$0 grid, then the largest guaranteed square grid shallow minor at depth $G\ast I_1\ast\dots\ast I_c-D$1 has side length

$G\ast I_1\ast\dots\ast I_c-D$2

with constants independent of $G\ast I_1\ast\dots\ast I_c-D$3, $G\ast I_1\ast\dots\ast I_c-D$4, and $G\ast I_1\ast\dots\ast I_c-D$5. The source’s typeset formula is malformed, but its proof and regime analysis identify the intended bound exactly in that form. The theorem is tight within constant factors, and it yields three regimes: linear-size shallow minors when damage is at most linear and depth is not restrictive, a genuinely shallow regime of order $G\ast I_1\ast\dots\ast I_c-D$6, and an unrestricted-minor regime of order $G\ast I_1\ast\dots\ast I_c-D$7 (Eppstein, 2013).

The same paper places shallow minors between two earlier guarantees. The largest undamaged square grid subgraph in an $G\ast I_1\ast\dots\ast I_c-D$8 grid with $G\ast I_1\ast\dots\ast I_c-D$9 deleted vertices has side length $I_i$0, using the exact lower bound

$I_i$1

For unrestricted minors, the guaranteed side length is

$I_i$2

more precisely

$I_i$3

The shallow-minor theorem is therefore a quantitative interpolation between the subgraph and ordinary-minor scales (Eppstein, 2013).

The lower-bound proofs are constructive. For very small damage, undamaged rows and columns already yield a large shallow grid minor, with bounded contraction radius verified by the spacing between consecutive intersections. In the depth-limited regime, the grid is partitioned into many subgrids so that one subgrid has at most $I_i$4 damaged vertices; the same rows-and-columns mechanism then applies locally. For sufficiently large $I_i$5, the ordinary minor construction can be made shallow by shrinking the chosen subgrids by a constant factor. The upper bound in the genuinely shallow regime uses evenly spaced diagonal damaged segments of length $I_i$6, planar duality, and a treewidth argument showing that every resulting shallow minor lives in a graph of treewidth $I_i$7 (Eppstein, 2013).

In higher dimensions, a lower bound persists. For a $I_i$8-dimensional cubical grid of side length $I_i$9 with $V_k$0 damaged vertices, there exists a depth-$V_k$1 $V_k$2-dimensional cubical grid shallow minor of side length

$V_k$3

The same source explicitly states that the corresponding upper bound is not known in higher dimensions, so the two-dimensional result is markedly sharper (Eppstein, 2013).

3. Excluding bounded-depth clique minors

A second major line of work studies graph classes that exclude a complete graph as a shallow minor. For integers $V_k$4 and $V_k$5, one paper denotes by

$V_k$6

the class of graphs excluding $V_k$7 as a minor of depth $V_k$8, and records the monotonicity

$V_k$9

In this setting, excluding a smaller depth is a stronger requirement (Wulff-Nilsen, 2014).

The structural consequence is a balanced-separator theorem of Plotkin, Rao, and Smith: if an $\lambda$0-vertex graph excludes $\lambda$1 as a depth $\lambda$2-minor, then it has a separator of size

$\lambda$3

Two later papers preserve this separator size while improving the algorithmic dependence on $\lambda$4, $\lambda$5, $\lambda$6, and $\lambda$7. One gives an

$\lambda$8

time algorithm that either produces a $\lambda$9-minor of depth $V_i$0 or outputs a separator of size

$V_i$1

Another gives three algorithms, including one with running time

$V_i$2

a second with

$V_i$3

when $V_i$4 and $V_i$5, and a third with

$V_i$6

in the same regime, at the cost of a larger separator

$V_i$7

All these algorithms are certificate-producing: they either return the separator or an explicit bounded-depth $V_i$8-minor model [(Wulff-Nilsen, 2011); (Wulff-Nilsen, 2014)].

The shared strategy is a separator-or-minor dichotomy. The generic framework maintains a partial clique-minor model by growing pairwise adjacent bounded-radius trees, and otherwise isolates a small BFS layer that can be added to the separator. In one version, the algorithm maintains a partition $V_i$9 and branch sets $\lambda$0 that certify a depth-$\lambda$1 $\lambda$2-minor; if $\lambda$3, the minor is complete, and if not, failure to extend the model yields a low-boundary set. The resulting depth guarantee comes from branch sets of diameter $\lambda$4 (Wulff-Nilsen, 2011).

The technical accelerations differ. One paper uses a dynamic constant-stretch spanner maintained under deletions, replacing repeated exact graph explorations by computations in the spanner; this is the source of the $\lambda$5 running time (Wulff-Nilsen, 2011). The later work uses decremental approximate distance oracles of Roditty and Zwick, with parameters

$\lambda$6

so that large approximate distance to an existing branch set certifies a genuinely far pair of vertices and triggers the separator step. Its faster implementations also use $\lambda$7-clusterings, nested $\lambda$8-clusterings, dense distance graphs, spanners, and mini-clusters. A key sparsity consequence there is that if

$\lambda$9

then any $0$0-vertex graph excluding $0$1 as a depth-$0$2 minor has only

$0$3

edges (Wulff-Nilsen, 2014).

4. Transfer to weak colouring numbers and graph products

The most direct connection between shallow minors and a formal “weak” parameter appears in work on graph products and beyond-planar classes. That paper uses the standard $0$4-shallow minor model with branch sets of radius at most $0$5, together with shallow topological minors and strong products. Its key transfer theorem states that if $0$6 is an $0$7-shallow minor of $0$8, then for every integer $0$9,

$H$0

Here the “weak” object is the weak colouring number $H$1, not a separately defined weak shallow minor (Hickingbotham et al., 2021).

The argument is path-lifting. If a vertex is strongly or weakly $H$2-reachable in the shallow minor, then tracing the corresponding branch sets in the host graph produces a witness path of length at most

$H$3

This converts bounded-radius minor models into quantitative bounds on reachability parameters. The paper also records the product estimate

$H$4

with an analogous bound for $H$5, and specializes it to $H$6 (Hickingbotham et al., 2021).

This transfer mechanism is then applied to several beyond-planar classes that are shown to be shallow minors, or shallow topological minors, of products involving planar or bounded-genus graphs and small complete graphs. The list includes powers of planar graphs, $H$7-planar graphs, $H$8-string graphs, fan-planar graphs, and $H$9-fan-bundle planar graphs. The resulting consequences are bounded queue-number, bounded nonrepetitive chromatic number, polynomial $r$00-centred chromatic numbers, linear strong colouring numbers, and cubic weak colouring numbers (Hickingbotham et al., 2021).

Several explicit examples are given. Every fan-planar graph is a $r$01-shallow minor of

$r$02

for some planar graph $r$03, which implies containment in

$r$04

for some graph $r$05 of treewidth at most $r$06, and therefore row treewidth at most $r$07. More generally, every $r$08-planar graph is a $r$09-shallow-topological minor of $r$10 for a graph $r$11 of Euler genus at most $r$12, and every $r$13-string graph is a $r$14-shallow minor of the same form (Hickingbotham et al., 2021).

The same paper also records a limitation. $r$15-gap planar graphs have at least exponential local treewidth and therefore cannot be described as a subgraph of the strong product of a graph with bounded treewidth and a path. This separates classes that admit shallow-minor/product control from classes that do not (Hickingbotham et al., 2021).

5. Hereditary analogues: shallow vertex minors

A distinct theory replaces contraction-based shallow minors by local-complementation-based shallow vertex minors. A depth-$r$16 vertex minor of a graph $r$17 is an induced subgraph of $r$18, where $r$19 is an independent set of $r$20; more generally, a depth-$r$21 shallow vertex minor has the form

$r$22

where each $r$23 is an independent set in the graph obtained after the previous rounds, and $r$24 is a set of deleted vertices. This depth parameter counts rounds of independent-set local complementation, not the radius or diameter of branch sets, so it is not a bounded-depth minor model in the usual shallow-minor sense (Buffière et al., 2024).

The conceptual role is analogous but the ambient category is different. Classical shallow minors are described there as the contraction-based notion underlying sparsity theory for monotone classes, whereas shallow vertex minors are introduced for hereditary classes. The paper proves two characterization theorems. A hereditary class $r$25 is dependent if and only if it does not contain all permutation graphs and, for each integer $r$26, excludes some split interval graph as a depth-$r$27 vertex minor. A hereditary class $r$28 is stable if and only if, for each integer $r$29, it excludes some half-graph as a depth-$r$30 vertex minor (Buffière et al., 2024).

The preservation statements are equally central. Stability is preserved under $r$31, hence under bounded-depth shallow vertex minors, and the same is true for dependence. The paper extends this preservation to hereditary classes of binary structures and deduces that bounded-depth shallow vertex minors of graphs with bounded twin-width again have bounded twin-width (Buffière et al., 2024).

The technical mechanism is a bounded-distortion lemma. If $r$32, then for vertices $r$33,

$r$34

A related statement shows that this distance preservation survives the introduction of bounded-complexity flips. Another structural lemma states that if $r$35 is a subdivision of $r$36 in which every edge is subdivided at most $r$37 times, then $r$38 is a depth-$r$39 vertex minor of $r$40 (Buffière et al., 2024). These are local-complementation analogues of the bounded-depth philosophy, but they do not define a weak shallow minor in the contraction-based sense.

6. Adjacent notions, hardness, and structural limits

Several further notions in the cited literature are close enough to cause terminological ambiguity. One paper studies dense $r$41-shallow topological minors and dense $r$42-subdivisions, rather than weak shallow minors. It defines an $r$43-subdivision as a subgraph in which every edge of a graph $r$44 is replaced by a path of exactly $r$45 edges, and an $r$46-shallow topological minor as a subgraph in which every edge is replaced by a path of length at most $r$47. These are already computationally hard: Dense $r$48-Shallow Topological Minor and Dense $r$49-Subdivision are NP-hard for $r$50, even on graphs that can be turned into subcubic planar graphs by deleting a single vertex, and for $r$51 they do not admit algorithms with running time

$r$52

unless ETH fails (Muzi et al., 2017).

Another paper introduces shallow-vortex minors, defined as minors of shallow-vortex grids $r$53. This is explicitly not the same notion as weak shallow minors or bounded-depth minors. Its relevance is surface-structural: for antichains $r$54 in the class $r$55, where members are non-planar Kuratowski-connected graphs and at least one member is a shallow-vortex minor, the paper proves that $r$56 has the Erdős-Pósa property in a minor-closed class $r$57 if and only if the parameter

$r$58

is bounded on $r$59. Here $r$60 is built from Dyck-grid families attached to minimal excluded surfaces. This is a surface-based shallow template, not a theory of weak shallow minors (Paul et al., 2024).

A different use of the word “weak” occurs in the Graph Minors weak structure theorem. An optimized version states that for every graphs $r$61 and $r$62, and every $r$63, either $r$64 is a minor of $r$65, or

$r$66

or there exists $r$67 with

$r$68

such that $r$69 contains a flat subdivided wall of height $r$70, whose compass has a rural division in which each internal flap has treewidth at most $r$71. For $r$72, the apex bound becomes $r$73, and the paper states that both the linear treewidth–wall relation and the number of deleted vertices are best possible (Giannopoulou et al., 2011). This is a weak structure theorem, not a theory of weak shallow minors, but it explains why large-treewidth minor-free graphs must contain a large planar-like local core.

The same terminological caution applies to weak $r$74-clustering. In the separator literature, weak $r$75-clustering means a clustering in which each cluster has at most $r$76 vertices and the total number of boundary vertices is $r$77; it is unrelated to weak shallow minors despite the similar adjective (Wulff-Nilsen, 2011).

Taken together, these works suggest a stable conceptual boundary. Standard shallow minors are the formal bounded-depth minor notion that carries most of the structural and algorithmic load. “Weak” enters nearby theories through weak colouring numbers, weak structure theorems, weak $r$78-clusterings, or distinct hereditary and surface analogues, but not through a single, uniformly adopted graph-theoretic notion of weak shallow minor [(Wulff-Nilsen, 2014); (Hickingbotham et al., 2021); (Buffière et al., 2024); (Paul et al., 2024); (Giannopoulou et al., 2011)].