H-Planarity and Parametric Extensions: when Modulators Act Globally

2025-07-11

AI Summary

This paper introduces the *H-Planarity* problem: given a hereditary, CMSO-definable, and polynomially decidable graph class ℋ, decide whether a graph G admits a vertex subset X—called a *planar ℋ-modulator*—such that every connected component of G − X belongs to ℋ and the *skeleton* of X (i.e., the quotient graph obtained by contracting each ℋ-component) is planar. To address this, we introduce two novel parameterized width measures: *ℋ-planar tree-depth* and *ℋ-planar tree-width*, generalizing classical elimination distance and treewidth to arbitrary graph classes ℋ. Leveraging motif counting, CMSO logic modeling, and dynamic programming on ℋ-planar decompositions, we design fixed-parameter tractable (FPT) algorithms. Our framework yields: (i) constant-factor approximation algorithms for graph coloring; (ii) polynomial-time algorithms for counting perfect matchings; and (iii) polynomial-time approximation schemes (PTAS) for problems such as Independent Set. The core contribution is a unified, ℋ-centric parameterized framework for planarity, enabling global structural control via motif-based decomposition.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Planning, Routing, and Scheduling: Activity and Plan Recognition
• Search and Optimization: Heuristic Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Develop polynomial-time algorithm for planar H-modulators in graphs
• Generalize elimination distance and tree decompositions to class H
• Provide FPT algorithms for graph problems using H-planar parameters

Innovation

• Polynomial-time algorithm for planar H-modulators
• Generalizes elimination distance with H-planar treedepth
• Extends tree decompositions via H-planar treewidth