  Bootstrap percolation is a deterministic cellular automaton in which vertices
of a graph~$G$ begin in one of two states, "dormant" or "active". Given a fixed
integer $r$, a dormant vertex becomes active if at any stage it has at least
$r$ active neighbors, and it remains active for the duration of the process.
Given an initial set of active vertices $A$, we say that $G$ $r$-percolates
(from $A$) if every vertex in $G$ becomes active after some number of steps.
Let $m(G,r)$ denote the minimum size of a set $A$ such that $G$ $r$-percolates
from $A$.
  Bootstrap percolation has been studied in a number of settings, and has
applications to both statistical physics and discrete epidemiology. Here, we
are concerned with degree-based density conditions that ensure $m(G,2)=2$. In
particular, we give an Ore-type degree sum result that states that if a graph
$G$ satisfies $\sigma_2(G)\ge n-2$, then either $m(G,2)=2$ or $G$ is in one of
a small number of classes of exceptional graphs. We also give a
Chv\'{a}tal-type degree condition: If $G$ is a graph with degree sequence
$d_1\le d_2\le\dots\le d_n$ such that $d_i \geq i+1$ or $d_{n-i} \geq n-i-1$
for all $1 \leq i < \frac{n}{2}$, then $m(G,2)=2$ or $G$ falls into one of
several specific exceptional classes of graphs. Both of these results are
inspired by, and extend, an Ore-type result in [D. Freund, M. Poloczek, and D.
Reichman, Contagious sets in dense graphs, to appear in European J. Combin.]
