On the size of immune sets in the k-PULL infection model

Abstract

This paper addresses immune sets in graphs, which are subsets of nodes that remain unaffected during the spread of influence, failure, or infection. The specific propagation model examined is the $k$-PULL infection rule, also referred to as bootstrap percolation. Studying immune sets offers important insights into the structural vulnerabilities and defensive capabilities of networks. In particular, we establish upper bounds for the size of minimal $k$-immune sets in graphs with a given maximum degree. Additionally, we focus on the $k$-immune number of a graph, defined as the minimum number of vertices in a $k$-immune set, and we derive bounds for this parameter. Lastly, we investigate the $k$-immune number of the Cartesian product of two graphs.