Equivalent Dichotomies for Triangle Detection in Subgraph, Induced, and Colored H-Free Graphs

2026-02-26

AI Summary

This work investigates when triangle detection becomes easier in graphs that exclude a fixed graph pattern $H$, and examines whether the computational complexity remains equivalent across three settings: subgraph, induced subgraph, and colored-graph formulations. By constructing a graph transformation $H^+$ that preserves both triangle count and 3-colorability, and by introducing a novel color-coding-inspired self-reduction technique, the authors reduce triangle detection in induced and colored $H$-free graphs to the standard $H^+$-free graph setting. This establishes, for the first time, the equivalence of tractability among all three models, yielding a unified dichotomy framework that provides a theoretical foundation for understanding subgraph detection complexity under forbidden pattern constraints.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Graph-based Machine Learning

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
• 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 recent paper by the authors (ITCS'26) initiates the study of the Triangle Detection problem in graphs avoiding a fixed pattern $H$ as a subgraph and proposes a \emph{dichotomy hypothesis} characterizing which patterns $H$ make the Triangle Detection problem easier in $H$-free graphs than in general graphs. In this work, we demonstrate that this hypothesis is, in fact, equivalent to analogous hypotheses in two broader settings that a priori seem significantly more challenging: \emph{induced} $H$-free graphs and \emph{colored} $H$-free graphs. Our main contribution is a reduction from the induced $H$-free case to the non-induced $\H^+$-free case, where $\H^+$ preserves the structural properties of $H$ that are relevant for the dichotomy, namely $3$-colorability and triangle count. A similar reduction is given for the colored case. A key technical ingredient is a self-reduction to Unique Triangle Detection that preserves the induced $H$-freeness property, via a new color-coding-like reduction.

Problem

• Triangle Detection
• H-free Graphs
• Induced Subgraphs
• Colored Graphs
• Dichotomy

Innovation

• dichotomy hypothesis
• induced H-free graphs
• colored H-free graphs
• self-reduction
• color-coding