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

TL;DR

This paper presents new graph and tree examples with non log-concave dominating set sequences generated using AI tools, and explores conditions under which these sequences are log-concave.

Contribution

It introduces AI-generated examples of non log-concave dominating set sequences and extends previous constructions to trees with multiple non log-concave indices.

Findings

Generated graphs and trees with non log-concave dominating set sequences.

Identified classes of caterpillar graphs with log-concave sequences.

Established a continuous analogue of the sequence that is always log-concave.

Abstract

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.