  Let $G_{n,p}^1$ be a superposition of the random graph $G_{n,p}$ and a
one-dimensional lattice: the $n$ vertices are set to be on a ring with fixed
edges between the consecutive vertices, and with random independent edges given
with probability $p$ between any pair of vertices. Bootstrap percolation on a
random graph is a process of spread of "activation" on a given realisation of
the graph with a given number of initially active nodes. At each step those
vertices which have not been active but have at least $r \geq 2$ active
neighbours become active as well. We study the size of the final active set in
the limit when $n\rightarrow \infty $. The parameters of the model are $n$, the
size $A_0=A_0(n)$ of the initially active set and the probability $p=p(n)$ of
the edges in the graph.
  Bootstrap percolation process on $G_{n,p}$ was studied earlier. Here we show
that the addition of $n$ local connections to the graph $G_{n,p}$ leads to a
more narrow critical window for the phase transition, preserving however, the
critical scaling of parameters known for the model on $G_{n,p}$. We discover a
range of parameters which yields percolation on $G_{n,p}^1$ but not on
$G_{n,p}$.
