We give new examples of graphs and trees with dominating set sequences that are not log-concave. These examples were generated by PatternBoost, a transformer-based reinforcement learning software developed by Charton-Ellenberg-Wagner-Williamson. We also show: for any positive integer $m$, there exists a tree whose dominating set sequence is not log-concave for at least $m$ indices by modifying a similar construction of Bautista-Ramos for the independent set sequence. We show that a large class of caterpillar graphs has log-concave dominating set sequences. A continuous analogue of the sequence is also log-concave for all graphs.

翻译：我们给出了控制集序列不满足对数凹性的图和树的新实例。这些实例由PatternBoost生成——这是一款由Charton、Ellenberg、Wagner和Williamson开发的基于Transformer的强化学习软件。我们还证明：对于任意正整数$m$，通过修改Bautista-Ramos针对独立集序列的类似构造，存在一棵树，其控制集序列至少在$m$个指标上不满足对数凹性。我们证明了一大类毛毛虫图具有对数凹的控制集序列。该序列的连续类比对于所有图也是对数凹的。