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

TL;DR

This paper establishes that the complexity dichotomy for Triangle Detection in $H$-free graphs extends equivalently to induced and colored $H$-free graphs, through reductions that preserve key structural properties.

Contribution

It proves the equivalence of the dichotomy hypothesis across subgraph, induced, and colored $H$-free graphs using novel reductions that maintain structural properties.

Findings

Reduces induced $H$-free case to non-induced $ ext{H}^+$-free case.

Provides a similar reduction for colored $H$-free graphs.

Introduces a color-coding-like reduction preserving induced $H$-freeness.

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.