Trees and Graphs with Non Log-concave Dominating Set Sequence via Patternboost

We study failures of log-concavity in dominating set sequences as a test case for AI-assisted mathematical discovery. We adapt PatternBoost, which alternates local search with transformer training on successful constructions, using objectives indexed relative to the domination number. The search reproduces known examples and finds additional graphs and trees, including a 32-vertex tree, with non-log-concave dominating set sequences. Additionally, by modifying a construction of Bautista-Ramos, we prove that for every positive integer $m$ there exists a tree with at least $m$ log-concavity violations. However, our Patternboost runs found no examples with multiple log-concavity violations. These results illustrate complementary roles for automated example generation and human mathematical construction, while leaving unimodality for trees unresolved.