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.

翻译：作者近期论文（ITCS'26）开创性地研究了在避免固定模式$H$作为子图的图中进行三角形检测的问题，并提出了一个\emph{二分性假设}，用于刻画哪些模式$H$会使$H$-自由图中的三角形检测问题比一般图中更容易。本工作证明，该假设实际上等价于两个先验上显著更具挑战性的更广泛场景中的类似假设：\emph{诱导}$H$-自由图与\emph{着色}$H$-自由图。我们的主要贡献是从诱导$H$-自由情形到非诱导$\H^+$-自由情形的归约，其中$\H^+$保留了与二分性相关的$H$的结构性质，即$3$-可着色性与三角形数量。针对着色情形给出了类似的归约。一个关键技术要素是通过新的类颜色编码归约，实现了在保持诱导$H$-自由性质的前提下向唯一三角形检测问题的自归约。