H-Planar Treewidth: A Hybrid Graph Parameter

• H-Planar treewidth is a hybrid parameter that decomposes a graph into a planar torso with bounded planar treewidth and additional H-class components.
• The framework employs tree and sphere decompositions to allow global modulators and manage large planar structures within the graph.
• This approach leads to fixed-parameter tractability with polynomial-time algorithms for recognition, coloring, and perfect matching in H-planar graphs.

to=arxiv_search 不中反json_string code 天天爱彩票怎么x? Let's see format. to=arxiv_search 天天种彩票ង់json code 天天中彩票不ార్క? to=arxiv_search 天天中彩票上json 凤凰大参考 code 平台开号{"^^^^^^^^2query2 and Parametric Extensions: when Modulators Act Globally^^^^^^^^2\2 to=search_arxiv 天天中彩票派奖json PRESERVED_PLACEHOLDER_^^^^2query2^^^^-planar treewidth is a hybrid width parameter introduced in the modulator/target framework of "H-Planarity and Parametric Extensions: when Modulators Act Globally" (&&&^^^^2query2^^^^&&&). It measures whether a graph can be organized around a torso of bounded planar treewidth while every connected component outside the modulator belongs to a fixed graph class PRESERVED_PLACEHOLDER_^^^^2\2^^^^. The parameter is paired with $H$-planarity and $H$-planar treedepth, and its defining feature is that the modulator may act globally: the remaining core may be an arbitrary planar graph and therefore need not have bounded ordinary treewidth.

^^^^2\2^^^^. Definition and formalism

For a graph $G$ and a vertex set $X \subseteq V(G)$, the torso of $X$ in $G$, denoted $\operatorname{torso}(G,X)$, is obtained from the induced subgraph $G[X]$ by turning PRESERVED_PLACEHOLDER_^^^^2\2query2^^^^ into a clique for every connected component PRESERVED_PLACEHOLDER_^^^^2\2\2^^^^ of PRESERVED_PLACEHOLDER_^^^^2\22^^^^. Equivalently, two vertices PRESERVED_PLACEHOLDER_^^^^2\2^^^^3 are adjacent in PRESERVED_PLACEHOLDER_^^^^2\2^^^^4 if either PRESERVED_PLACEHOLDER_^^^^2\2^^^^5, or some connected component of PRESERVED_PLACEHOLDER_^^^^2\2^^^^6 is adjacent to both PRESERVED_PLACEHOLDER_^^^^2\2^^^^7 and PRESERVED_PLACEHOLDER_^^^^2\2^^^^8 (&&&^^^^2query2^^^^&&&).

Given a graph class PRESERVED_PLACEHOLDER_^^^^2\2^^^^9, a set $H$^^^^2query2^^^^ is a planar $H$^^^^2\2^^^^-modulator if $H$2 is planar and every connected component of $H$3 belongs to $H$4. A graph is $H$5-planar if it admits such a modulator. The associated decision problem is $H$6-PLANARITY: given $H$7, decide whether $H$8 admits a planar $H$9-modulator.

The paper places this inside a general operator scheme. For a union-closed graph class $H$^^^^2query2^^^^ and a minor-monotone parameter $H$^^^^2\2^^^^,

$H$2

Here $H$3 is interpreted componentwise because $H$4 is union-closed. In this language, $H$5-planar treewidth is $H$6, where $H$7 denotes planar treewidth.

Planar treewidth itself is defined as follows. The planar treewidth of a graph $H$8, denoted $H$9, is the minimum $G$^^^^2query2^^^^ such that $G$^^^^2\2^^^^ admits a tree decomposition in which each bag either has size at most $G$2 or has a planar torso. Consequently,

$G$3

With $G$4 denoting the class of graphs of planar treewidth at most $G$5, the notation $G$6 denotes exactly the class of graphs of $G$7-planar treewidth at most $G$8. This operator viewpoint is central: $G$9-planar treewidth is not an isolated definition, but one instance of a general “torso parameter plus target class” construction.

2. Decomposition viewpoint

The intended decomposition-theoretic meaning is explicit. A graph has $X \subseteq V(G)$^^^^2query2^^^^-planar treewidth at most $X \subseteq V(G)$^^^^2\2^^^^ if it admits a tree decomposition such that every leaf-bag belongs to $X \subseteq V(G)$2, and all other bags are either of size at most $X \subseteq V(G)$3 or have a planar torso (&&&^^^^2query2^^^^&&&). This is the analogue of $X \subseteq V(G)$4-treewidth, but with planar torsos allowed in place of uniformly bounded-width bags.

The paper develops a second formulation through sphere decompositions. In Section 6 it defines a local cell property $X \subseteq V(G)$5: roughly, after cliquifying the boundary of a cell, the resulting graph must admit a $X \subseteq V(G)$6-modulator containing that boundary. If a graph has a sphere decomposition all of whose cells satisfy $X \subseteq V(G)$7, then the graph has $X \subseteq V(G)$8-planar treewidth at most $X \subseteq V(G)$9. Conversely, on the unbreakable instances used in the recognition algorithm, bounded $X$^^^^2query2^^^^-planar treewidth implies the existence of such sphere decompositions.

This decomposition viewpoint is paired with $X$^^^^2\2^^^^-planar treedepth. Planar treedepth $X$2 is defined recursively: $X$3 if $X$4 is empty, and $X$5 if there exists $X$6 with $X$7 planar and $X$8. Then $X$9-planar treedepth is $G$^^^^2query2^^^^. The pair $G$^^^^2\2^^^^ provides recursive and decomposition-based extensions of $G$2-planarity.

A basic normalization follows immediately from the definition: planar graphs have planar treewidth $G$3, since a single bag containing all vertices has planar torso. This is one reason the framework is described as a parametric extension of planarity rather than merely another bounded-width formalism.

3. Relation to adjacent width notions

The paper states that both $G$4-planar treedepth and $G$5-planar treewidth are more general than $G$6-treedepth or $G$7-treewidth (&&&^^^^2query2^^^^&&&). The reason is structural: planar torsos may have arbitrarily large ordinary treewidth, so bounded $G$8-planar treewidth does not force bounded $G$9-treewidth.

It also differs from deletion distance. On highly connected graphs, however, the distinction collapses: if $\operatorname{torso}(G,X)$^^^^2query2^^^^ is $\operatorname{torso}(G,X)$^^^^2\2^^^^-connected, then a graph $\operatorname{torso}(G,X)$2 can be obtained from $\operatorname{torso}(G,X)$3 by at most $\operatorname{torso}(G,X)$4 vertex deletions if and only if the $\operatorname{torso}(G,X)$5-planar treewidth of $\operatorname{torso}(G,X)$6 is at most $\operatorname{torso}(G,X)$7. Thus, in high connectivity, the hybrid parameter becomes equivalent to a deletion parameter.

A common source of confusion is the distinction between $\operatorname{torso}(G,X)$8-planar treewidth and planar-treewidth. In "5-Approximation for $\operatorname{torso}(G,X)$9-Treewidth Essentially as Fast as $G[X]$^^^^2query2^^^^-Deletion Parameterized by Solution Size" (Jansen et al., 2023), planar-treewidth means $G[X]$^^^^2\2^^^^-treewidth for $G[X]$2: large planar induced subgraphs are placed in leaf-local base components, and only the nonplanar interfaces count toward the width. By contrast, $G[X]$3-planar treewidth is defined through planar torsos and components in $G[X]$4; its controlling object is a planar torso inside the modulator/torso formalism, not a planar base component in a tree $G[X]$5-decomposition.

A second nearby notion is the existence of tree decompositions whose bags themselves induce low-treewidth graphs. "Optimal Tree-Decompositions with Bags of Bounded Treewidth" proves that every planar graph has an optimal tree-decomposition in which every bag has treewidth at most $G[X]$6 (Hendrey et al., 27 Nov 2025). That result is adjacent in spirit, but it concerns induced bag subgraphs at optimal decomposition width, whereas $G[X]$7-planar treewidth is defined by planar torsos and $G[X]$8-components. This suggests that $G[X]$9-planar treewidth belongs to a wider family of planar-hybrid decomposition parameters, but it is not reducible to bounded induced bag treewidth.

4. Recognition and parameterized tractability

The first algorithmic layer is PRESERVED_PLACEHOLDER_^^^^2\2query2query2^^^^-PLANARITY itself. If PRESERVED_PLACEHOLDER_^^^^2\2query2\2^^^^ is hereditary, CMSO-definable, and decidable in polynomial time, then PRESERVED_PLACEHOLDER_^^^^2\2query22^^^^-PLANARITY is solvable in polynomial time (&&&^^^^2query2^^^^&&&). This gives polynomial-time recognition of graphs that admit a planar PRESERVED_PLACEHOLDER_^^^^2\2query2^^^^3-modulator.

The main parameterized transfer theorem concerns PRESERVED_PLACEHOLDER_^^^^2\2query2^^^^4-planar treedepth and PRESERVED_PLACEHOLDER_^^^^2\2query2^^^^5-planar treewidth. Let PRESERVED_PLACEHOLDER_^^^^2\2query2^^^^6, and let PRESERVED_PLACEHOLDER_^^^^2\2query2^^^^7 be hereditary, CMSO-definable, and union-closed. If PRESERVED_PLACEHOLDER_^^^^2\2query2^^^^8-DELETION parameterized by solution size PRESERVED_PLACEHOLDER_^^^^2\2query2^^^^9 is solvable in time PRESERVED_PLACEHOLDER_^^^^2\2\2query2^^^^, then there is a non-uniform algorithm deciding whether PRESERVED_PLACEHOLDER_^^^^2\2\2\2^^^^ in time

PRESERVED_PLACEHOLDER_^^^^2\2\22^^^^

For PRESERVED_PLACEHOLDER_^^^^2\2\2^^^^3, this yields fixed-parameter tractability of bounded PRESERVED_PLACEHOLDER_^^^^2\2\2^^^^4-planar treewidth from fixed-parameter tractability of deletion into PRESERVED_PLACEHOLDER_^^^^2\2\2^^^^5.

A central restricted-instance theorem underlies this transfer. For hereditary union-closed PRESERVED_PLACEHOLDER_^^^^2\2\2^^^^6, and integers PRESERVED_PLACEHOLDER_^^^^2\2\2^^^^7, there is an algorithm that, given an PRESERVED_PLACEHOLDER_^^^^2\2\2^^^^8-unbreakable graph PRESERVED_PLACEHOLDER_^^^^2\2\2^^^^9, checks whether PRESERVED_PLACEHOLDER_^^^^2\22query2^^^^ has PRESERVED_PLACEHOLDER_^^^^2\22\2^^^^-planar treewidth at most PRESERVED_PLACEHOLDER_^^^^2\222^^^^ in time

PRESERVED_PLACEHOLDER_^^^^2\22^^^^3

The general theorem then follows from the CMSO reduction to unbreakable graphs.

The framework is also constructive at the decomposition level. Under the assumptions of the main theorem, if PRESERVED_PLACEHOLDER_^^^^2\22^^^^4 has PRESERVED_PLACEHOLDER_^^^^2\22^^^^5-planar treewidth at most PRESERVED_PLACEHOLDER_^^^^2\22^^^^6, one can construct a PRESERVED_PLACEHOLDER_^^^^2\22^^^^7-modulator PRESERVED_PLACEHOLDER_^^^^2\22^^^^8 together with a tree decomposition of PRESERVED_PLACEHOLDER_^^^^2\22^^^^9 of planar width at most PRESERVED_PLACEHOLDER_^^^^2\2max_results2query2^^^^.

5. Structural techniques and global modulators

The technical novelty of the framework is the treatment of modulators that act globally. Earlier modulator/treewidth meta-theorems typically require modulators of low bidimensionality, so that the modulator does not spread throughout a large wall or grid minor. Here the planar torso itself may spread across the whole wall, and this breaks the usual locality-based irrelevant-vertex strategy (&&&^^^^2query2^^^^&&&).

To overcome this, the paper combines planar PRESERVED_PLACEHOLDER_^^^^2\2max_results2\2^^^^-modulators with sphere decompositions, well-linked and ground-maximal renditions, and a new irrelevant-vertex argument. Large apex grids serve as obstructions, while flat walls are used positively. In the wall case, one removes a central vertex and proves irrelevance by comparing decompositions inside the compass with decompositions outside the central region.

For PRESERVED_PLACEHOLDER_^^^^2\2^^^^32-planar treewidth, the key irrelevance statement is Lemma ^^^^2\2^^^^7. Let PRESERVED_PLACEHOLDER_^^^^2\2^^^^33 be hereditary, let PRESERVED_PLACEHOLDER_^^^^2\2^^^^34 be an PRESERVED_PLACEHOLDER_^^^^2\2^^^^35-unbreakable graph, let PRESERVED_PLACEHOLDER_^^^^2\2^^^^36 be a flatness pair of height PRESERVED_PLACEHOLDER_^^^^2\2^^^^37, and let PRESERVED_PLACEHOLDER_^^^^2\2^^^^38 be a central vertex of PRESERVED_PLACEHOLDER_^^^^2\2^^^^39, where PRESERVED_PLACEHOLDER_^^^^2\2H-Planarity and Parametric Extensions: when Modulators Act Globally2query2^^^^ is odd. Then PRESERVED_PLACEHOLDER_^^^^2\2H-Planarity and Parametric Extensions: when Modulators Act Globally2\2^^^^ has PRESERVED_PLACEHOLDER_^^^^2\2^^^^42-planar treewidth at most PRESERVED_PLACEHOLDER_^^^^2\2^^^^43 if and only if both the PRESERVED_PLACEHOLDER_^^^^2\2^^^^44-compass of PRESERVED_PLACEHOLDER_^^^^2\2^^^^45 and PRESERVED_PLACEHOLDER_^^^^2\2^^^^46 have PRESERVED_PLACEHOLDER_^^^^2\2^^^^47-planar treewidth at most PRESERVED_PLACEHOLDER_^^^^2\2^^^^48. This is the analogue of the classical irrelevant-vertex lemma, redesigned for globally acting modulators.

Planarity of the torso is not a cosmetic condition. It is used to bound attachment neighborhoods, to obtain sphere embeddings and renditions, to invoke flat-wall structure, and to make the gluing of local decompositions topologically consistent. The paper also notes a hardness barrier in the background: without the planarity requirement on the torso, the corresponding partition problem becomes NP-hard even for PRESERVED_PLACEHOLDER_^^^^2\2^^^^49.

6. Applications, examples, and limitations

The framework applies to graph classes PRESERVED_PLACEHOLDER_^^^^2\25-Approximation for $\operatorname{torso}(G,X)$9-Treewidth Essentially as Fast as $G[X]$0-Deletion Parameterized by Solution Size2query2^^^^ satisfying the theorem assumptions, and the paper explicitly points to examples such as edgeless graphs, bipartite graphs, and perfect graphs (&&&^^^^2query2^^^^&&&). It also identifies target classes on which the downstream applications are meaningful, including bounded clique-width classes, classes excluding a shallow-vortex minor, chain graphs, co-chain graphs, threshold graphs, perfect graphs, and PRESERVED_PLACEHOLDER_^^^^2\25-Approximation for $\operatorname{torso}(G,X)$9-Treewidth Essentially as Fast as $G[X]$0-Deletion Parameterized by Solution Size2\2^^^^-free graphs.

For coloring, if PRESERVED_PLACEHOLDER_^^^^2\2^^^^52 is hereditary, CMSO-definable, polynomial-time decidable, and PRESERVED_PLACEHOLDER_^^^^2\2^^^^53 is polynomial-time computable on graphs in PRESERVED_PLACEHOLDER_^^^^2\2^^^^54, then every PRESERVED_PLACEHOLDER_^^^^2\2^^^^55-planar graph can be colored in polynomial time with at most PRESERVED_PLACEHOLDER_^^^^2\2^^^^56 colors. In the bounded-parameter setting, if PRESERVED_PLACEHOLDER_^^^^2\2^^^^57, then PRESERVED_PLACEHOLDER_^^^^2\2^^^^58 can be colored in polynomial time using at most

PRESERVED_PLACEHOLDER_^^^^2\2^^^^59

colors.

For counting perfect matchings, if PRESERVED_PLACEHOLDER_^^^^2\2Optimal Tree-Decompositions with Bags of Bounded Treewidth2query2^^^^ is hereditary, CMSO-definable, polynomial-time decidable, and counting weighted or unweighted perfect matchings is polynomial-time solvable on graphs in PRESERVED_PLACEHOLDER_^^^^2\2Optimal Tree-Decompositions with Bags of Bounded Treewidth2\2^^^^, then the same counting problem is polynomial-time solvable on PRESERVED_PLACEHOLDER_^^^^2\2^^^^62-planar graphs. If PRESERVED_PLACEHOLDER_^^^^2\2^^^^63, the weighted or unweighted number of perfect matchings can be computed in time

PRESERVED_PLACEHOLDER_^^^^2\2^^^^64

For Maximum Independent Set, if PRESERVED_PLACEHOLDER_^^^^2\2^^^^65 is hereditary, CMSO-definable, polynomial-time decidable, and Maximum Independent Set is polynomial-time solvable on PRESERVED_PLACEHOLDER_^^^^2\2^^^^66, then there is an algorithm that, given PRESERVED_PLACEHOLDER_^^^^2\2^^^^67 and an PRESERVED_PLACEHOLDER_^^^^2\2^^^^68-planar graph PRESERVED_PLACEHOLDER_^^^^2\2^^^^69, computes in time

PRESERVED_PLACEHOLDER_^^^^2\2^^^^7^^^^2query2^^^^

an independent set of size at least PRESERVED_PLACEHOLDER_^^^^2\2^^^^7^^^^2\2 The paper presents this as an EPTAS consequence of the planar-torso framework.

The main limitations are also explicit. The recognition theorems are non-uniform because they rely on the CMSO-to-unbreakable reduction theorem. Heredity is essential: PRESERVED_PLACEHOLDER_^^^^2\2^^^^72-PLANARITY can be NP-hard without it, and the paper proves NP-hardness even for PRESERVED_PLACEHOLDER_^^^^2\2^^^^73. CMSO-definability is part of the current theorem statement, but whether it is necessary is left open.

In summary, PRESERVED_PLACEHOLDER_^^^^2\2^^^^74-planar treewidth is a hybrid parameter in which the controlling torso may be planar rather than bounded-treewidth, and the components outside the modulator belong to PRESERVED_PLACEHOLDER_^^^^2\2^^^^75. Its significance lies in making “global” planar modulators algorithmically tractable while preserving access to tree-decomposition methods. The resulting theory sits between planarity, PRESERVED_PLACEHOLDER_^^^^2\2^^^^76-deletion, and hybrid-width parameters, and it supports fixed-parameter recognition together with concrete algorithms for coloring, counting perfect matchings, and approximation schemes.