--- title: H-Planar Treedepth Analysis url: https://www.emergentmind.com/topics/h-planar-treedepth type: topic --- # H-Planar Treedepth Analysis Searching arXiv for the cited paper and closely related foundational work on elimination distance / treedepth parametrizations. tool call: arxiv_search(query="2507.08541 OR \"H-Planarity and Parametric Extensions: when Modulators Act Globally\"", max_results=5) tool result: { "results": [ { "arxiv_id": "2507.08541", "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": "2025-07-11", "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 $H$, 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" [2507.08541], 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 $N_G(C)\cap X$ into a clique. Fix a target graph class $H$. A set $X \subseteq V(G)$ is a planar $H$-modulator if two conditions hold: $\operatorname{torso}(G,X)$ is planar, and every connected component of $G-X$ belongs to $H$. The associated decision problem, H-PLANARITY, asks whether $G$ admits such a set $X$ [2507.08541]. For union-closed $H$, the $H$-planar treedepth of $G$, denoted $\operatorname{ptd}_H(G)$, is defined by $$ \operatorname{ptd}_H(G)=\min \left\{ k \;\middle|\; \exists X_1,\dots,X_k \subseteq V(G)\ \text{s.t. for each } i,\ \operatorname{torso}\!\left(G-(X_1\cup\cdots\cup X_{i-1}),X_i\right)\ \text{is planar, and every component of } G-(X_1\cup\cdots\cup X_k)\ \text{lies in } H \right\}. $$ Equivalently, one removes $k$ planar layers so that the remainder is in $H$. A recurring misconception is to read $\operatorname{ptd}_H$ 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 $H$. ## 2. Relation to treedepth, elimination distance, and the modulator framework Standard treedepth is defined recursively by $$ \operatorname{td}(G)=0 \text{ if } G \text{ is empty,} $$ and otherwise $$ \operatorname{td}(G)=1+\min_{v\in V(G)} \operatorname{td}(G-v). $$ Equivalently, one may speak of an elimination sequence picking one vertex per step. Elimination distance to $H$, denoted $H$-td$(G)$, is given by $$ H\text{-td}(G)=0 \iff G\in H, $$ and $$ H\text{-td}(G)\le k \iff \exists X\subseteq V(G)\ \text{with } |X|=1 \text{ such that } \operatorname{torso}(G,X)\in H \text{ and } H\text{-td}(G-X)\le k-1. $$ This generalizes $\operatorname{td}$ by taking $H=\{\text{edgeless graphs}\}$. Planar treedepth $\operatorname{ptd}(G)$ is presented as the special case where $H=\{\text{empty graph}\}$: 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 $p$ such as size, $\operatorname{td}$, $\operatorname{tw}$, $\operatorname{ptd}$, or $\operatorname{ptw}$, and any target class $H$, one sets $$ H\text{--}p(G)=\min_{X\subseteq V(G)} \{\, p(\operatorname{torso}(G,X)) \mid G-X\in H \,\}. $$ Within this template, $p=\text{size}$ yields classical H-DELETION, $p=\operatorname{td}$ yields elimination distance to $H$, and $p=\operatorname{ptd}$ 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 $H$. ## 3. Decidability and fixed-parameter tractability The foundational algorithmic statement is the polynomial-time decidability of H-PLANARITY. If $H$ is hereditary, CMSO-definable, and polynomial-time decidable, then H-PLANARITY can be decided in time $n^{O(1)}$ [2507.08541]. The parametric extension to H-planar treedepth is an FPT result. If $H$ is hereditary, CMSO-definable, union-closed, and polynomial-time decidable, and if H-DELETION is FPT in $k$, then deciding whether $\operatorname{ptd}_H(G)\le k$ is FPT in $k$, with running time $f(k)\cdot n^{4+o(1)}$. 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$(k)$-PLANARITY for $k=O(1)$. Third, H$(k)$-PLANARITY is FPT in $k$ by a new irrelevant-vertex technique on large flat walls. For the H-planar treedepth parameter itself, the paper states that once a planar $H$-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 $H$-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 $H$-modulator [2507.08541]. The self-reduction proceeds via forbidden subgraphs. One first finds a minimal obstacle $F\notin H$ with $|F|=O_H(1)$. Then, for each vertex $v$, one tests whether “$G$ plus a gadget attached at $v$” is H-planar; the procedure keeps $v$ in $X$ exactly if the test succeeds. This yields, in polynomial time, a planar $H$-modulator $X$ whenever one exists. This constructive step matters because H-planar treedepth is defined through successive planar layers, not only through existential quantification. Once $X$ 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 $G-X$ is summarized by a clique on its neighborhood in $X$, 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 $G$ excludes a big clique-minor, or has bounded treewidth, or contains a large flat wall [2507.08541]. For H$(k)$-planarity, the paper states an irrelevant vertex lemma on a large flat wall. In a large flat wall, one finds a vertex $v$ whose removal preserves the existence of a planar H$(k)$-modulator. More precisely, a large flat wall $W$ certified by $(A,B,P,C,\pi)$ enables the implication that if $G-v$ and the $A$-compass are both H$(k)$-planar, then so is $G$. The proof sketch relies on constructing two compatible sphere-decompositions, one for $G-v$ and one for the compass, and gluing them along the track of a mid-layer of $W$. 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 $\operatorname{ptd}_H(G)\le k$, then one either finds a $K_{4k}$-minor or a small apex set $A$ and a large flat wall in $G-A$. One then peels off one central vertex $v$ and recurses on $G-v$. 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 $H=\{\text{edgeless graphs}\}$ and states that then H-planar treedepth is planar treedepth of $G$ [2507.08541]. Let $G$ be a cycle $C_6$ with a path $P_3$ attached at one vertex. Removing $X_1=\{\text{central vertex of }P_3\}$ gives a planar torso, and the remainder is an edgeless graph consisting of two isolated vertices. Thus $\operatorname{ptd}_H(G)\le 1$. The paper further states that $\operatorname{td}(G)=3$ but planar treedepth improves it to $1$. 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 $H$. For graph coloring, the paper gives an additive $4$-approximation on any H-planar graph. One finds a planar $H$-modulator $X$, colors $\operatorname{torso}(G,X)$ with $4$ colors by the Four-Color Theorem, and colors each $H$-component optimally in polynomial time by assumption. The total number of colors is at most $\chi(G-X)+4=\chi(G)+4$. For counting perfect matchings, the method uses the modulator $X$ to carve $G$ into planar flaps plus small $H$-blocks. Valiant’s matchgate gadgets are then applied recursively across small separators of size at most $3$, eventually reducing the problem to counting in planar graphs by FKT and in $H$ 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 $X$. In $\operatorname{torso}(G,X)$, one performs an $r$-layering around an arbitrary root in $O(1/\varepsilon)$ 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 $c(\varepsilon)\cdot n^{O(1)}$, then picks the best solution and argues a $(1-\varepsilon)$-approximation. The paper states that these applications extend mutatis mutandis to the parametric cases $\operatorname{ptd}_H(G)\le k$ and H-planar treewidth once one has a certifying elimination sequence or H-tree-decomposition, via Corollaries 9 and 10. 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. Source: https://www.emergentmind.com/topics/h-planar-treedepth