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

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 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: induced H-free graphs and 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.