Equivalent Dichotomies for Triangle Detection in Subgraph, Induced, and Colored H-Free Graphs
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arXiv
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| Formato: | Preprint |
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2026
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| _version_ | 1866910034180440064 |
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| author | Abboud, Amir Safier, Ron Wallheimer, Nathan |
| author_facet | Abboud, Amir Safier, Ron Wallheimer, Nathan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_23196 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Equivalent Dichotomies for Triangle Detection in Subgraph, Induced, and Colored H-Free Graphs Abboud, Amir Safier, Ron Wallheimer, Nathan Data Structures and Algorithms Computational Complexity Combinatorics 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. |
| title | Equivalent Dichotomies for Triangle Detection in Subgraph, Induced, and Colored H-Free Graphs |
| topic | Data Structures and Algorithms Computational Complexity Combinatorics |
| url | https://arxiv.org/abs/2602.23196 |