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.

We initiate the study of combinatorial algorithms for Triangle Detection in H-free graphs. The goal is to decide if a graph that forbids a fixed pattern H as a subgraph contains a triangle, using only "combinatorial" methods that notably exclude fast matrix multiplication. Our work aims to classify which patterns admit a subcubic speedup, working towards a dichotomy theorem. On the lower bound side, we show that if H is not 3-colorable or contains more than one triangle, the complexity of the problem remains unchanged, and no combinatorial speedup is likely possible. On the upper bound side, we develop an embedding approach that results in a strongly subcubic, combinatorial algorithm for a rich class of "embeddable" patterns. Specifically, for an embeddable pattern of size k, our algorithm runs in Õ(n^{3-1/(2^{k-3)}}) time, where Õ(⋅) hides poly-logarithmic factors. This algorithm also extends to listing all the triangles within the same time bound. We supplement this main result with two generalizations: - A generalization to patterns that are embeddable up to a single obstacle that arises from a triangle in the pattern. This completes our classification for small patterns, yielding a dichotomy theorem for all patterns of size up to eight. - An H-sensitive algorithm for embeddable patterns, which runs faster when the number of copies of H is significantly smaller than the maximum possible Ω(n^{k}). Finally, we focus on the special case of odd cycles. We present specialized Triangle Detection algorithms that are very efficient: - A combinatorial algorithm for C_{2k+1}-free graphs that runs in Õ(m+n^{1+2/k}) time for every k ≥ 2, where m is the number of edges in the graph. - A combinatorial C₅-sensitive algorithm that runs in Õ(n² + n^{4/3} t^{1/3}) time, where t is the number of 5-cycles in the graph.

Sumsets are central objects in additive combinatorics. In 2007, Granville asked whether one can efficiently recognize whether a given set S is a sumset, i.e. whether there is a set A such that A + A = S. Granville suggested an algorithm that takes exponential time in the size of the given set, but can we do polynomial or even linear time? This basic computational question is indirectly asking a fundamental structural question: do the special characteristics of sumsets allow them to be efficiently recognizable? In this paper, we answer this question negatively by proving that the problem is NP-complete. Specifically, our results hold for integer sets and over any finite field. Assuming the Exponential Time Hypothesis, our lower bound becomes

This is a survey of the exciting recent progress made in understanding the complexity of distributed subgraph finding problems. It overviews the results and techniques for assorted variants of subgraph finding problems in various models of distributed computing, and states intriguing open questions. This version contains some updates over the ICALP 2021 version, and I will try to keep updating it as additional progress is made.