  We provide an explicit algorithm for sampling a uniform simple connected
random graph with a given degree sequence. By products of this central result
include:
  (i) continuum scaling limits of uniform simple connected graphs with given
degree sequence and asymptotics for the number of simple connected graphs with
given degree sequence under some regularity conditions, and
  (ii) scaling limits for the metric space structure of the maximal components
in the critical regime of both the configuration model and the uniform simple
random graph model with prescribed degree sequence under finite third moment
assumption on the degree sequence.
  As a substantive application we answer a question raised by Cerny and
Teixeira by obtaining the metric space scaling limit of maximal components in
the vacant set left by random walks on random regular graphs.
