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

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.

25 pages, 11 figures, 4 tables