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

2026-05-03

AI Summary

This study challenges the prevailing assumption that domination polynomials of graphs and trees are universally log-concave. By integrating a Transformer-based reinforcement learning framework (PatternBoost), combinatorial construction techniques, and continuous analog analysis, the authors present the first automatically generated examples of graphs and trees whose domination polynomials violate log-concavity. The work establishes two main results: first, for any positive integer \( m \), there exists a tree whose domination polynomial exhibits at least \( m \) consecutive non-log-concave coefficients; second, it identifies a class of caterpillar graphs whose domination polynomials are provably log-concave, a property further confirmed in their continuous analogues. These findings advance the constructive theory of non-log-concave domination polynomials and delineate precise boundary conditions for this combinatorial property.

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.

Problem

• dominating set sequence
• log-concave
• trees
• graphs
• combinatorics

Innovation

• PatternBoost
• dominating set sequence
• log-concavity
• caterpillar graphs
• transformer-based reinforcement learning