H-Planarity: Planar Torsos and Global Modulators

• H-Planarity is defined by the existence of a planar torso from a vertex set whose deletion leaves components in a target graph class H.
• The framework innovatively uses global modulators that allow nonlocal spread while preserving planarity through clique closures in the torso.
• Extensions like H-planar treedepth and H-planar treewidth enable efficient algorithms, leveraging CMSO techniques and bounded component strategies.

Searching arXiv for papers on H-Planarity and closely related planarity notions. H-Planarity denotes, in its recent explicit formulation, the problem of deciding whether a graph $G$ admits a planar $\mathcal H$-modulator: a vertex set $X\subseteq V(G)$ such that the torso of $X$ is planar and every connected component of $G-X$ belongs to a target graph class $\mathcal H$. Introduced as a decomposition paradigm in 2025, the notion treats planarity as the “torso side” of a modulator/target scheme and extends naturally to the parameters $\mathcal H$-planar treedepth and $\mathcal H$-planar treewidth. The same label has also been used informally for several distinct $H$-indexed planarity questions in neighboring literatures, so the current usage is best understood as a specific structural and algorithmic notion rather than a universal term (Fomin et al., 11 Jul 2025).

1. Definition and structural core

For a graph $G$ and a set $\mathcal H$0, the torso $\mathcal H$1 is obtained from $\mathcal H$2 by making a clique out of the neighborhood in $\mathcal H$3 of every connected component of $\mathcal H$4. Equivalently, if $\mathcal H$5 is the set of connected components of $\mathcal H$6, then for each $\mathcal H$7 all pairs of vertices in $\mathcal H$8 are joined by edges. A set $\mathcal H$9 is a planar $X\subseteq V(G)$0-modulator if $X\subseteq V(G)$1 is planar and every connected component of $X\subseteq V(G)$2 belongs to $X\subseteq V(G)$3. A graph with such a set is called $X\subseteq V(G)$4-planar (Fomin et al., 11 Jul 2025).

The associated decision problem is $X\subseteq V(G)$5-PLANARITY: given $X\subseteq V(G)$6, determine whether $X\subseteq V(G)$7 admits a planar $X\subseteq V(G)$8-modulator. The paper packages this in a class-composition operator $X\subseteq V(G)$9: for graph classes $X$0, the class $X$1 consists of graphs having a set $X$2 whose torso lies in $X$3 and whose deleted components all lie in $X$4. Thus $X$5, where $X$6 is the class of planar graphs, is exactly the class of $X$7-planar graphs. The operator is associative, and the iterates $X$8 with $X$9 are used later for recursive parameters (Fomin et al., 11 Jul 2025).

The torso condition is essential. A planar induced subgraph $G-X$0 does not capture the global effect of the deleted components, because each deleted component can connect multiple vertices of $G-X$1 and thereby simulate missing edges in the planar remainder. This suggests why the framework differs sharply from ordinary vertex-deletion to planarity: the “kept side” is not merely planar as an induced subgraph, but planar after recording external attachments as clique closures.

2. Why the modulator is global

A central theme of the theory is that these modulators can act globally. In classical modulator frameworks, the modulator often has bounded treewidth or otherwise behaves locally relative to large walls or grid minors. Here the torso is only required to be planar, and planar graphs may have arbitrarily large treewidth. As a result, the modulator need not be localized in a bounded-width region and may spread through the graph in a genuinely nonlocal way (Fomin et al., 11 Jul 2025).

This distinction is visible in the basic separator behavior. If $G-X$2 excludes $G-X$3 as a minor and $G-X$4 is a $G-X$5-modulator, then every component $G-X$6 of $G-X$7 satisfies

$G-X$8

For $G-X$9, this gives $\mathcal H$0. The bound is small, but it does not imply that the planar torso itself has bounded width. That is precisely why older irrelevant-vertex paradigms based on bounded-treewidth torsos do not apply directly (Fomin et al., 11 Jul 2025).

The framework therefore occupies an intermediate position between deletion to a target class and decomposition over bounded-width torsos. It preserves enough of planarity to inherit planar algorithmics, while allowing the retained structural core to be much richer than a bounded-treewidth graph.

3. Polynomial-time solvability of $\mathcal H$1-PLANARITY

The main theorem states that if $\mathcal H$2 is hereditary, CMSO-definable, and decidable in polynomial time, then $\mathcal H$3-PLANARITY is solvable in polynomial time. The proof is non-constructive: it uses the CMSO meta-theorem of Lokshtanov, Ramanujan, Saurabh, and Zehavi to reduce the problem to highly unbreakable graphs, and that reduction is itself non-constructive in the sense that it proves existence of a polynomial-time algorithm without directly yielding a uniform implementation (Fomin et al., 11 Jul 2025).

The algorithmic core is a bounded-component version. For $\mathcal H$4, let $\mathcal H$5 denote the graphs in $\mathcal H$6 with at most $\mathcal H$7 vertices. The paper proves an explicit algorithm for $\mathcal H$8-PLANARITY running in time

$\mathcal H$9

for some computable function $\mathcal H$0. This result is combined with the CMSO reduction and a separate treatment of the case where one deleted component is large (Fomin et al., 11 Jul 2025).

The proof uses a flat-wall trichotomy. Given $\mathcal H$1 and parameters $\mathcal H$2, one can either find an apex grid minor of height $\mathcal H$3, bound the treewidth by $\mathcal H$4 with

$\mathcal H$5

or obtain a flatness pair $\mathcal H$6 whose compass has treewidth at most $\mathcal H$7. If $\mathcal H$8 contains an apex grid $\mathcal H$9 with

$\mathcal H$0

then $\mathcal H$1 has no planar $\mathcal H$2-modulator. If the treewidth is bounded, Courcelle’s theorem applies because $\mathcal H$3 is finite and hence CMSO-definable. The difficult case is a large flat wall with bounded-treewidth compass, where the proof establishes an irrelevant vertex rule for a central wall vertex (Fomin et al., 11 Jul 2025).

The technical novelty lies in sphere decompositions. The paper proves that $\mathcal H$4 is $\mathcal H$5-planar if and only if it has an $\mathcal H$6-compatible sphere decomposition. It then compares two types of such decompositions, called ground-maximal and well-linked, and shows that a ground-maximal rendition is more grounded than a well-linked rendition of the same society. This comparison enables a gluing theorem across a tracked cycle in the flat wall and yields the equivalence

$\mathcal H$7

where $\mathcal H$8 is the wall compass and $\mathcal H$9 is a central wall vertex. That equivalence is the relevant irrelevant-vertex rule (Fomin et al., 11 Jul 2025).

The assumptions are substantive. Heredity is essential: for the non-hereditary class $H$0, $H$1-PLANARITY is NP-complete. By contrast, CMSO-definability is required by the proof technique, but the paper explicitly does not claim it is necessary in principle (Fomin et al., 11 Jul 2025).

4. Parametric extensions

Two recursive parameters extend the basic notion. The first is planar treedepth $H$2, defined by $H$3 for the empty graph and, for $H$4,

$H$5

if there exists $H$6 such that $H$7 is planar and $H$8. The class of graphs of planar treedepth at most $H$9 is denoted $G$0, and graphs of $G$1-planar treedepth at most $G$2 are exactly those in $G$3 (Fomin et al., 11 Jul 2025).

The second is planar treewidth $G$4. A graph has $G$5 if it admits a tree decomposition such that every bag either has size at most $G$6 or has a planar torso. The $G$7-parametric version, $G$8-planar treewidth, generalizes $G$9-treewidth by allowing planar torsos in internal bags rather than only bounded-size bags. The abstract summarizes it as follows: a graph has $\mathcal H$00-planar treewidth at most $\mathcal H$01 if it admits a tree decomposition such that every leaf-bag is contained in $\mathcal H$02, and all other bags are either of size at most $\mathcal H$03 or have a planar torso (Fomin et al., 11 Jul 2025).

These parameters fit a general scheme. For a minor-monotone parameter $\mathcal H$04 and a union-closed graph class $\mathcal H$05,

$\mathcal H$06

Taking $\mathcal H$07 gives elimination distance to $\mathcal H$08; taking $\mathcal H$09 gives $\mathcal H$10-treewidth; taking $\mathcal H$11 yields the new parameters (Fomin et al., 11 Jul 2025).

The main fixed-parameter theorem states that if $\mathcal H$12 is hereditary, CMSO-definable, and union-closed, and if $\mathcal H$13-DELETION is FPT parameterized by solution size $\mathcal H$14 with running time $\mathcal H$15, then for $\mathcal H$16 there is a non-uniform algorithm deciding $\mathcal H$17 in time

$\mathcal H$18

Thus many existing deletion-to-$\mathcal H$19 algorithms lift automatically to the planar-torso parameters (Fomin et al., 11 Jul 2025).

5. Other meanings of “H-Planarity” in the literature

The terminology is not uniform. Several older or neighboring lines of work use “H-Planarity” informally for distinct $\mathcal H$20-indexed planarity questions.

Interpretation	Core question	Representative paper
Planar $\mathcal H$21-modulator	Does $\mathcal H$22 admit $\mathcal H$23 with planar torso and components of $\mathcal H$24 in $\mathcal H$25?	(Fomin et al., 11 Jul 2025)
Planar covering with fixed target $\mathcal H$26	Does a planar graph $\mathcal H$27 cover a fixed graph $\mathcal H$28 via a locally bijective homomorphism?	(Bílka et al., 2011)
Partial embedding extension of $\mathcal H$29	Can a prescribed planar embedding of a subgraph $\mathcal H$30 be extended to all of $\mathcal H$31?	(Fink et al., 2024)
Hierarchical embedding constraints	Does $\mathcal H$32 admit a planar embedding whose cyclic orders satisfy FPQ-tree constraints?	(Liotta et al., 2019)

In planar covering, the objects are a fixed target graph $\mathcal H$33 and an input planar graph $\mathcal H$34. The problem $\mathcal H$35 asks whether $\mathcal H$36 covers $\mathcal H$37, meaning that there exists a homomorphism $\mathcal H$38 that is locally bijective on closed neighborhoods. For several small targets, including $\mathcal H$39, $\mathcal H$40, $\mathcal H$41, $\mathcal H$42, $\mathcal H$43, and the dumbbell multigraph $\mathcal H$44, the paper proves NP-completeness of $\mathcal H$45 (Bílka et al., 2011).

In partially embedded planarity, the input is $\mathcal H$46, where $\mathcal H$47 and $\mathcal H$48 is a prescribed planar embedding of $\mathcal H$49. The task is to decide whether $\mathcal H$50 extends to a planar embedding of $\mathcal H$51. A 2024 paper gives a linear-time algorithm based on modified PC-trees and the Booth–Lueker vertex-addition test, explicitly treating this as the standard formalization of what many authors informally call $\mathcal H$52-planarity in the partial-embedding sense (Fink et al., 2024).

In hierarchical constrained planarity, each vertex is equipped with one or more FPQ-trees specifying allowed cyclic orders of incident edges. The problem, called FPQ-choosable Planarity Testing, is fixed-parameter tractable for biconnected graphs when parameterized by treewidth and the number of FPQ-trees per vertex, NP-complete when parameterized only by the number of FPQ-trees, and W[1]-hard when parameterized only by treewidth (Liotta et al., 2019).

Several neighboring models sit close to this terminological field without using the same definition. $\mathcal H$53-planarity studies hybrid drawings with clusters of size at most $\mathcal H$54 and at most $\mathcal H$55 boundary ports per vertex; $\mathcal H$56-Clique2Path Planarity asks whether each designated clique can be reduced to a path so that the resulting graph is planar; and 4-embeddability asks whether a planar graph $\mathcal H$57 is a subgraph of some 4-regular planar graph, decidable in $\mathcal H$58 time (Giacomo et al., 2018, Angelini et al., 2018, Dowden et al., 2011).

6. Applications, boundaries, and significance

The 2025 theory is not only existential. Although the main polynomial-time theorem is non-constructive, the paper derives a polynomial-time self-reduction that constructs a planar $\mathcal H$59-modulator whenever one exists. Analogous constructive corollaries recover certifying elimination sequences for bounded $\mathcal H$60-planar treedepth and suitable decompositions for bounded $\mathcal H$61-planar treewidth (Fomin et al., 11 Jul 2025).

Once such a decomposition is known, several algorithmic consequences follow. If $\mathcal H$62 is hereditary, CMSO-definable, polynomial-time decidable, and optimally colorable in polynomial time, then every $\mathcal H$63-planar graph can be colored in polynomial time with at most $\mathcal H$64 colors. If $\mathcal H$65-ptd$\mathcal H$66, the bound becomes $\mathcal H$67; if $\mathcal H$68-ptw$\mathcal H$69, it becomes $\mathcal H$70. Under analogous assumptions for perfect matching counting on $\mathcal H$71, weighted and unweighted perfect matchings are polynomial-time computable on $\mathcal H$72-planar graphs, and for $\mathcal H$73-ptw$\mathcal H$74 the running time is $\mathcal H$75. If Maximum Independent Set is polynomial-time solvable on $\mathcal H$76, then there is an EPTAS for Maximum Independent Set on $\mathcal H$77-planar graphs with running time $\mathcal H$78 (Fomin et al., 11 Jul 2025).

Representative target classes include bipartite graphs, perfect graphs, bounded clique-width graphs, graphs excluding a shallow-vortex as a minor, chain graphs, co-chain graphs, threshold graphs, and $\mathcal H$79-free graphs, depending on the downstream algorithmic task. This suggests that H-Planarity is best viewed as a transfer principle: it imports planar-graph algorithmics into mixed structures whose nonplanar parts are confined to components from $\mathcal H$80, while the torso retains global planar control (Fomin et al., 11 Jul 2025).

The framework also has sharp boundaries. The paper notes that if one only required $\mathcal H$81 to be planar, rather than $\mathcal H$82, the problem becomes NP-hard by partition hardness results, even for hereditary disjoint-union-closed $\mathcal H$83. This clarifies the conceptual role of the torso: it is the device that makes the decomposition both structurally faithful and algorithmically tractable (Fomin et al., 11 Jul 2025).

In its current sense, H-Planarity is therefore a theory of planar torsos and global modulators. Its significance lies less in a single recognition problem than in the decomposition template it provides: a graph can be algorithmically close to planar not because a small set of vertices restores planarity, but because the entire nonplanar remainder decomposes into $\mathcal H$84-components whose effect on the retained core can still be summarized by a planar torso.