H-Planar Treedepth Analysis

• H-planar treedepth is defined as the minimum number of planar layers to remove so that the remaining components belong to a target graph class H.
• It generalizes classical treedepth and elimination distance by integrating a modulator/target framework with planar torso constructions.
• The parameter leads to efficient FPT algorithms and offers practical applications in graph coloring, counting perfect matchings, and independent set approximations.

Searching arXiv for the cited paper and closely related foundational work on elimination distance / treedepth parametrizations. tool call: arxiv_search(query="^^^^^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^^^^^"H-Planarity and Parametric Extensions: when Modulators Act Globally\"", max_results=5) tool result: { "results": [ { "arxiv_id": "(&&&^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^&&&)", "title": "H-Planarity and Parametric Extensions: when Modulators Act Globally", "authors": [ "Marthe Bonamy", "Michał Pilipczuk", "Paweł Rzążewski", "Michał Wrochna" ], "abstract": "We introduce a series of graph decompositions based on the modulator/target scheme of modification problems that enable several algorithmic applications that parametrically extend the algorithmic potential of planarity. In the core of our approach is a polynomial time algorithm for computing planar H-modulators. Given a graph class H, a planar H-modulator of a graph G is a set X \subseteq V(G) such that the ``torso'' of X is planar and all connected components of G - X belong to H. Here, the torso of X is obtained from G[X] if, for every connected component of G-X, we form a clique out of its neighborhood on G[X]. We introduce H-Planarity as the problem of deciding whether a graph G has a planar H-modulator. We prove that, if H is hereditary, CMSO-definable, and decidable in polynomial time, then H-Planarity is solvable in polynomial time. Further, we introduce two parametric extensions of H-Planarity by defining the notions of H-planar treedepth and H-planar treewidth, which generalize the concepts of elimination distance and tree decompositions to the class H. Combining this result with existing FPT algorithms for various H-modulator problems, we thereby obtain FPT algorithms parameterized by H-planar treedepth and H-planar treewidth for numerous graph classes H. By combining the well-known algorithmic properties of planar graphs and graphs of bounded treewidth, our methods for computing H-planar treedepth and H-planar treewidth lead to a variety of algorithmic applications. For instance, once we know that a given graph has bounded H-planar treedepth or bounded H-planar treewidth, we can derive additive approximation algorithms for graph coloring and polynomial-time algorithms for counting (weighted) perfect matchings. Furthermore, we design Efficient Polynomial-Time Approximation Schemes (EPTAS-es) for several problems, including Maximum Independent Set.", "published": "2^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^25-^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^7-11", "categories": [ "cs.DM" ] } ] } H-planar treedepth is a parameter introduced as a parametric extension of H-Planarity that combines planar modulators with elimination-style depth. For a union-closed target graph class PRESERVED_PLACEHOLDER_^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^, it measures the minimum number of planar layers whose removal leaves a graph whose remaining connected components all belong to $H$. In the formulation of "H-Planarity and Parametric Extensions: when Modulators Act Globally" (&&&^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^&&&), the parameter sits inside a broader modulator/target scheme that also captures classical deletion problems, elimination distance, and planar treewidth analogues.

1. Formal definition

The underlying object is the torso. For a finite, undirected graph $G$ and a set $X \subseteq V(G)$, the torso of $X$ in $G$, written $\operatorname{torso}(G,X)$, is obtained from the induced subgraph $G[X]$ by, for each connected component $C$ of $G-X$, turning $H$^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^ into a clique.

Fix a target graph class $H$1. A set $H$2 is a planar $H$3-modulator if two conditions hold: $H$4 is planar, and every connected component of $H$5 belongs to $H$6. The associated decision problem, H-PLANARITY, asks whether $H$7 admits such a set $H$8 (&&&^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^&&&).

For union-closed $H$9, the $G$^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^-planar treedepth of $G$1, denoted $G$2, is defined by

$G$3

Equivalently, one removes $G$4 planar layers so that the remainder is in $G$5.

A recurring misconception is to read $G$6 as a plain deletion depth. The definition is stricter: at each stage, the removed layer must induce a planar torso in the current residual graph, and only after all layers are removed may the residual graph decompose into components from $G$7.

2. Relation to treedepth, elimination distance, and the modulator framework

Standard treedepth is defined recursively by

$G$8

and otherwise

$G$9

Equivalently, one may speak of an elimination sequence picking one vertex per step.

Elimination distance to $X \subseteq V(G)$^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^, denoted $X \subseteq V(G)$1-td$X \subseteq V(G)$2, is given by

$X \subseteq V(G)$3

and

$X \subseteq V(G)$4

This generalizes $X \subseteq V(G)$5 by taking $X \subseteq V(G)$6.

Planar treedepth $X \subseteq V(G)$7 is presented as the special case where $X \subseteq V(G)$8: one layer removes a planar modulator whose torso is planar. H-planar treedepth combines these two directions of generalization.

The paper places these notions in a uniform modulator/measure template. For any minor-monotone graph parameter $X \subseteq V(G)$9 such as size, $X$^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^, $X$1, $X$2, or $X$3, and any target class $X$4, one sets

$X$5

Within this template, $X$6 yields classical H-DELETION, $X$7 yields elimination distance to $X$8, and $X$9 yields H-planar treedepth. This suggests that H-planar treedepth is not an isolated parameter but part of a systematic way to transfer algorithmic properties from a target structure on the torso to a residual class $G$^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^.

3. Decidability and fixed-parameter tractability

The foundational algorithmic statement is the polynomial-time decidability of H-PLANARITY. If $G$1 is hereditary, CMSO-definable, and polynomial-time decidable, then H-PLANARITY can be decided in time $G$2 (&&&^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^&&&).

The parametric extension to H-planar treedepth is an FPT result. If $G$3 is hereditary, CMSO-definable, union-closed, and polynomial-time decidable, and if H-DELETION is FPT in $G$4, then deciding whether $G$5 is FPT in $G$6, with running time $G$7. The same statement holds for H-planar treewidth.

The proof outline for H-PLANARITY uses three ingredients. First, a meta-theorem of Lokshtanov et al. reduces any CMSO-problem to the unbreakable case. Second, two subcases are handled in polynomial time: the BIG-LEAF case, which finds small separators by brute force, and the SMALL-LEAF case, which reduces to H$G$8-PLANARITY for $G$9. Third, H$\operatorname{torso}(G,X)$^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^-PLANARITY is FPT in $\operatorname{torso}(G,X)$1 by a new irrelevant-vertex technique on large flat walls.

For the H-planar treedepth parameter itself, the paper states that once a planar $\operatorname{torso}(G,X)$2-modulator is available, one can peel off one layer at a time for treedepth, or build a decomposition for treewidth, invoking the H-PLANARITY algorithm on smaller instances or on the torso subgraph. The computational content is therefore both recognitional and constructive.

4. Constructing planar $\operatorname{torso}(G,X)$3-modulators

Although the polynomial-time theorem for H-PLANARITY is described as nonconstructive, the paper gives a polynomial-time self-reduction for actually building a planar $\operatorname{torso}(G,X)$4-modulator (&&&^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^&&&).

The self-reduction proceeds via forbidden subgraphs. One first finds a minimal obstacle $\operatorname{torso}(G,X)$5 with $\operatorname{torso}(G,X)$6. Then, for each vertex $\operatorname{torso}(G,X)$7, one tests whether “$\operatorname{torso}(G,X)$8 plus a gadget attached at $\operatorname{torso}(G,X)$9” is H-planar; the procedure keeps $G[X]$^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^ in $G[X]$1 exactly if the test succeeds. This yields, in polynomial time, a planar $G[X]$2-modulator $G[X]$3 whenever one exists.

This constructive step matters because H-planar treedepth is defined through successive planar layers, not only through existential quantification. Once $G[X]$4 is in hand, the algorithm can recurse on smaller instances and certify each layer explicitly. The same infrastructure underlies the H-planar treewidth construction, where the output is a decomposition rather than an elimination sequence.

The torso viewpoint is central throughout. Each component of $G[X]$5 is summarized by a clique on its neighborhood in $G[X]$6, so the interaction between the modulator and the residual graph is encoded globally rather than locally. This suggests why the paper frames the setting as one in which modulators act globally.

5. Structural theorems: flat walls, irrelevance, and recursion

The main structural input is the Flat Wall Theorem, attributed in the paper to Robertson–Seymour and Kawarabayashi–Thomas–Wollan. Its role is described as follows: either $G[X]$7 excludes a big clique-minor, or has bounded treewidth, or contains a large flat wall (&&&^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^&&&).

For H$G[X]$8-planarity, the paper states an irrelevant vertex lemma on a large flat wall. In a large flat wall, one finds a vertex $G[X]$9 whose removal preserves the existence of a planar H$C$^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^-modulator. More precisely, a large flat wall $C$1 certified by $C$2 enables the implication that if $C$3 and the $C$4-compass are both H$C$5-planar, then so is $C$6.

The proof sketch relies on constructing two compatible sphere-decompositions, one for $C$7 and one for the compass, and gluing them along the track of a mid-layer of $C$8. Ground-maximality and well-linkedness ensure that no “crossing” cells spoil planarity.

For planar H-treedepth, the paper gives a separate sketch. One replaces the big flat-wall irrelevance argument by the classical Flat Wall Theorem plus a layering argument. If $C$9, then one either finds a $G-X$^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^-minor or a small apex set $G-X$1 and a large flat wall in $G-X$2. One then peels off one central vertex $G-X$3 and recurses on $G-X$4. The H-planar treewidth argument is described as very similar, except that one builds a tree decomposition instead of an elimination sequence. In both cases, the flat-wall machinery provides the structural bottleneck through which the FPT algorithm proceeds.

6. Example and algorithmic consequences

The paper includes an illustrative example with $G-X$5 and states that then H-planar treedepth is planar treedepth of $G-X$6 (&&&^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^&&&). Let $G-X$7 be a cycle $G-X$8 with a path $G-X$9 attached at one vertex. Removing $H$^^^^1(Fomin et al., 11 Jul 2025) OR \1(Fomin et al., 11 Jul 2025) OR \1^^^^ gives a planar torso, and the remainder is an edgeless graph consisting of two isolated vertices. Thus $H$^^^^1(Fomin et al., 11 Jul 2025) OR \11^^^^. The paper further states that $H$^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^2 but planar treedepth improves it to $H$^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^3.

The algorithmic applications exploit the fact that bounded H-planar treedepth and bounded H-planar treewidth combine algorithmic properties of planar graphs with those of graphs controlled by $H$^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^4.

For graph coloring, the paper gives an additive $H$^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^5-approximation on any H-planar graph. One finds a planar $H$^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^6-modulator $H$^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^7, colors $H$^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^8 with $H$^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^9 colors by the Four-Color Theorem, and colors each $H$1^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^-component optimally in polynomial time by assumption. The total number of colors is at most $H$11.

For counting perfect matchings, the method uses the modulator $H$12 to carve $H$13 into planar flaps plus small $H$14-blocks. Valiant’s matchgate gadgets are then applied recursively across small separators of size at most $H$15, eventually reducing the problem to counting in planar graphs by FKT and in $H$16 by assumption. The abstract states the broader conclusion as polynomial-time algorithms for counting (weighted) perfect matchings.

For Maximum Independent Set, the paper gives an EPTAS. Using Theorem 8, one first finds $H$17. In $H$18, one performs an $H$19-layering around an arbitrary root in $H$2^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^ layers. Each subgraph with one residue class of layers removed has bounded local H-treewidth, since the torso minus one layer is planar. On each such subgraph, one dynamic-programs on its H-treewidth decomposition in time $H$21, then picks the best solution and argues a $H$22-approximation.

The paper states that these applications extend mutatis mutandis to the parametric cases $H$23 and H-planar treewidth once one has a certifying elimination sequence or H-tree-decomposition, via Corollaries 9 and 1^^^^1(Fomin et al., 11 Jul 2025) OR \1^^^^. A plausible implication is that H-planar treedepth functions not only as a structural measure but also as an algorithmic interface: once a graph is certified to have bounded depth in this sense, approximation, counting, and decomposition-based dynamic programming become available in a unified manner.