  We show that the extended based mapping class group of an infinite-type
surface is naturally isomorphic to the automorphism group of the loop graph of
that surface. Additionally, we show that the extended mapping class group
stabilizing a finite set of punctures is isomorphic to the arc graph relative
to that finite set of punctures. This extends a known result for sufficiently
complex finite-type surfaces, and provides a new angle from which to study the
mapping class groups of infinite-type surfaces.
