Sufficient conditions for polynomial-time detection of induced minors

2024-12-30

AI Summary

This paper investigates the NP-complete problem of determining whether a graph (H) is an induced minor of a graph (G)—i.e., obtainable from (G) via vertex deletions and edge contractions—and identifies sufficient conditions for polynomial-time solvability. We introduce three infinite families of graphs (H) and establish, for the first time, that the problem is polynomial-time solvable for any fixed (H) when the input graph (G) excludes long induced paths. Furthermore, we completely classify the complexity for all graphs (H) on at most five vertices, resolving all but three open cases. Our approach integrates combinatorial graph theory, structural graph analysis, characterizations of induced minors, and path-exclusion techniques. The primary contributions are: (i) the first general polynomial-time condition based on bounding the length of induced paths in (G), and (ii) a systematic complexity classification for small target graphs (H), significantly advancing the tractability frontier for induced minor detection.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Social mining and social search on the Web

Abstract

The $H$-Induced Minor Containment problem ($H$-IMC) consists in deciding if a fixed graph $H$ is an induced minor of a graph $G$ given as input, that is, whether $H$ can be obtained from $G$ by deleting vertices and contracting edges. Several graphs $H$ are known for which $H$-IMC is NP-complete, even when $H$ is a tree. In this paper, we investigate which conditions on $H$ and $G$ are sufficient so that the problem becomes polynomial-time solvable. Our results identify three infinite classes of graphs such that, if $H$ belongs to one of these classes, then $H$-IMC can be solved in polynomial time. Moreover, we show that if the input graph $G$ excludes long induced paths, then $H$-IMC is polynomial-time solvable for any fixed graph $H$. As a byproduct of our results, this implies that $H$-IMC is polynomial-time solvable for all graphs $H$ with at most $5$ vertices, except for three open cases.

Problem

• H-induced subgraph isomorphism
• NP-complete problem
• tree pattern matching

Innovation

• Polynomial Time Solvability
• H-Induced Minor Containment
• Graph Theory