H-Planarity and Parametric Extensions: when Modulators Act Globally

TL;DR

This work introduces H-planarity, a framework that augments planarity with a target class H through a planar H-modulator whose modulator-torso is planar and whose outside components lie in H. It proves polynomial-time solvability when H is hereditary, CMSO-definable, and poly-time decidable, and extends the framework to parametric extensions via H-planar treedepth (ptd) and H-planar treewidth (ptw), yielding FPT algorithms under suitable conditions. The authors develop a non-uniform Irrelevant Vertex technique adapted to globally modifying modulators, leveraging flat walls, renditions, sphere decompositions, and ground-maximal/well-linked concepts to glue local solutions into global witnesses. They demonstrate broad algorithmic applications, including additive coloring bounds, polynomial-time counting of weighted perfect matchings, and EPTAS for Independent Set, across various H-structured graph classes. Overall, the work bridges planarity with structural graph parameters, enabling tractable algorithms for a wide class of modification problems and suggesting a pathway to extending these techniques to broader minor-monotone parameters.

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.

Figures (16)

• Figure 1: Example of an $\mathcal{H}$-planar graph $G$. The vertices of the $\mathcal{H}$-modulator $X$ and the edges of $G[X]$ are black, the edges of the torso of $X$ that are not edges of $G$ are shown in dashed red, and the connected components of $G-X$ together with the edges joining them with $X$ are colored blue.
• Figure 2: Illustration of a graph with $\mathcal{H}$-planar treedepth at most three. The sequence $X_1,X_2,X_3$ is a certifying elimination sequence.
• Figure 3: A rendition.
• Figure 4: A wall.
• Figure 5: An $\mathcal{H}$-compatible cell $c$ (in gray). The $\mathcal{H}$-modulator $X$ is the set of red and black vertices, with the black vertices being the vertices of $\tilde{c}$. The blue, green, and purple balls are the connected components obtained after removing $X$, and the dashed lines are used for the edges of the torso that are not necessarily edges of the graph.

Theorems & Definitions (109)

• Theorem 1
• Proposition 1: AgrawalKLPRSZ22
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Proposition 2: Courcelle's Theorem Courcelle90themArnborgLS91easy
• Definition 1: Unbreakable graph
• Proposition 3: Theorem 1, LokshtanovR0Z18redu
• Lemma 1