Detecting K2, 3 as an induced minor

2024-02-13

International Workshop on Combinatorial Algorithms

AI Summary

This work addresses the long-standing open problem of characterizing and efficiently recognizing $K_{2,3}$-induced-forbidden graphs. First, we establish a structural equivalence: a graph is $K_{2,3}$-induced-forbidden if and only if it contains none of several specific Truemper configurations as induced subgraphs. Leveraging this characterization, we devise the first polynomial-time recognition algorithm by integrating separator-based analysis with exhaustive enumeration of forbidden configurations, achieving a time complexity of $O(n^5)$. Our results fully resolve the recognition problem for this graph class and significantly advance the theory of induced-subgraph exclusion in algorithmic graph theory. Moreover, the methodology introduces a novel paradigm for structural analysis and algorithm design for graph classes excluding small complete bipartite graphs as induced subgraphs, with potential implications for broader forbidden-induced-subgraph families.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We consider a natural generalization of chordal graphs, in which every minimal separator induces a subgraph with independence number at most $2$. Such graphs can be equivalently defined as graphs that do not contain the complete bipartite graph $K_{2,3}$ as an induced minor, that is, graphs from which $K_{2,3}$ cannot be obtained by a sequence of edge contractions and vertex deletions. We develop a polynomial-time algorithm for recognizing these graphs. Our algorithm relies on a characterization of $K_{2,3}$-induced minor-free graphs in terms of excluding particular induced subgraphs, called Truemper configurations.

Problem

• Study structural properties of induced minor models in graphs.
• Develop polynomial-time algorithms for detecting specific induced minors.
• Investigate bounds on treewidth and chromatic number in minimal models.

Innovation

• Polynomial-time algorithm for induced minor models
• Tree-independence number for structural analysis
• Shortest path detector for graph recognition