  We study a model for the destruction of a random network by fire. Suppose
that we are given a multigraph of minimum degree at least 2 having real-valued
edge-lengths. We pick a uniform point from along the length and set it alight;
the edges of the multigraph burn at speed 1. If the fire reaches a vertex of
degree 2, the fire gets directly passed on to the neighbouring edge; a vertex
of degree at least 3, however, passes the fire either to all of its neighbours
or none, each with probability $1/2$. If the fire goes out before the whole
network is burnt, we again set fire to a uniform point. We are interested in
the number of fires which must be set in order to burn the whole network, and
the number of points which are burnt from two different directions. We analyse
these quantities for a random multigraph having $n$ vertices of degree 3 and
$\alpha(n)$ vertices of degree 4, where $\alpha(n)/n \to 0$ as $n \to \infty$,
with i.i.d. standard exponential edge-lengths. Depending on whether $\alpha(n)
\gg \sqrt{n}$ or $\alpha(n)=O(\sqrt{n})$, we prove that as $n \to \infty$ these
quantities converge jointly in distribution when suitably rescaled to either a
pair of constants or to (complicated) functionals of Brownian motion.
  We use our analysis of this model to make progress towards a conjecture of
Aronson, Frieze and Pittel concerning the number of vertices which remain
unmatched when we use the Karp-Sipser algorithm to find a matching on the
Erd\H{o}s-R\'enyi random graph.
