Minimum Weakly Saturated Graphs and Bootstrap Percolation in General Host Graphs

A graph $G$ is weakly $H$-saturated if one can obtain $K_n$ by adding one edge to $G$ at a time, where each additional edge creates at least one new copy of $H$. The minimum number of edges needed for a weakly $H$-saturated graph $G$ of order $n$ is known as the weak saturation number of $H$, written $wsat(n,H)$. A graph $G$ is minimum weakly saturated if $wsat(n,G)=\mathrm{\mid }E(G)\mathrm{\mid }-1$ for some value of $n$. We explore classes of minimum weakly saturated graphs and their connection to the $H$-bootstrap percolation process, as well as weak saturation in a more general setting than the complete graph.