# Irreversible $k$-Threshold Dynamics on Corona and Base-$b$ Corona Product Graphs

## TL;DR

This paper analyzes the irreversible $k$-threshold process on corona-type graph products, providing exact conversion numbers, a unified framework for base-$b$ constructions, and probabilistic insights into activation likelihoods.

## Contribution

It introduces exact results for the $k$-threshold conversion number on corona and double corona graphs, and extends the analysis to a generalized base-$b$ corona framework.

## Key findings

- Exact $k$-threshold conversion numbers for corona and double corona graphs.
- A unified framework for base-$b$ corona constructions.
- Probabilistic analysis of seed set effectiveness.

## Abstract

We study the irreversible $k$-threshold process on corona-type graph products, including the corona product, the double corona product, and a generalized base-$b$ corona construction. Exact results are obtained for the irreversible $k$-threshold conversion number on corona and double corona product graphs through reduction lemmas that relate these graphs to smaller corona-type instances and to classical base graphs. The base-$b$ construction provides a unified framework extending the corona and double corona cases. A probabilistic refinement is also considered by studying the likelihood that a uniformly chosen minimum seed set yields complete activation. These results show how layered attachment structure influences both deterministic conversion behavior and probabilistic saturation in corona-type graph products.