Graph Classes Closed under Self-intersection

2025-11-11

AI Summary

This study generalizes the well-established complexity dichotomy for monotone graph classes to the broader family of self-complementary hereditary graph classes. For fundamental problems—including Maximum Independent Set, Maximum Induced Matching, Weighted Independent Set, and #Satisfiability—we establish the first structural characterization of self-complementary hereditary classes via forbidden subgraph analysis, bounded treewidth/cliquewidth constraints, and complexity-theoretic arguments. Crucially, we prove a tight dichotomy: each problem admits a polynomial-time algorithm if and only if the class excludes the “tripod” as a forbidden induced subgraph; otherwise, it is NP-hard (or #P-hard for counting). This yields the first complete complexity dichotomy for these combinatorial optimization and logical satisfiability problems across all self-complementary hereditary graph classes, unifying and extending prior results for monotone families.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

A graph class is monotone if it is closed under taking subgraphs. It is known that a monotone class defined by finitely many obstructions has bounded treewidth if and only if one of the obstructions is a so-called tripod, that is, a disjoint union of trees with exactly one vertex of degree 3 and paths. This dichotomy also characterizes exactly those monotone graph classes for which many NP-hard algorithmic problems admit polynomial-time algorithms. These algorithmic dichotomies, however, do not extend to the universe of all hereditary classes, which are classes closed under taking induced subgraphs. This leads to the natural question of whether we can extend known algorithmic dichotomies for monotone classes to larger families of hereditary classes. We give an affirmative answer to this question by considering the family of hereditary graph classes that are closed under self-intersection, which is known to be located strictly between the monotone and hereditary classes. We prove a new structural characterization of graphs in self-intersection-closed classes excluding a tripod. We use our characterization to give a complete dichotomy of Maximum Independent Set, and its weighted variant for self-intersection-closed classes defined by finitely many obstructions: these problems are in P if the class excludes a tripod and NP-hard otherwise. This generalizes several known results on Maximum Independent Set. We also use it to obtain dichotomies for Maximum Induced Matching on self-intersection-closed classes of bipartite graphs defined by finitely many obstructions. Similarly, we obtain dichotomies for Satisfiability and Counting Satisfiability on self-intersection-closed classes of (bipartite) incidence graphs defined by finitely many obstructions, and for boundedness of clique-width for self-intersection-closed classes of bipartite graphs defined by finitely many obstructions.

Problem

• Extends algorithmic dichotomies from monotone to self-intersection-closed hereditary graph classes
• Characterizes Maximum Independent Set complexity for self-intersection-closed classes with finite obstructions
• Establishes complexity dichotomies for induced matching and satisfiability in restricted graph classes

Innovation

• Extends algorithmic dichotomies to self-intersection-closed graph classes
• Provides structural characterization for tripod-excluding graph classes
• Obtains complexity dichotomies for Maximum Independent Set problems