  We describe a new method for the random sampling of connected networks with a
specified degree sequence. We consider both the case of simple graphs and that
of loopless multigraphs. The constraints of fixed degrees and of connectedness
are two of the most commonly needed ones when constructing null models for the
practical analysis of physical or biological networks. Yet handling these
constraints, let alone combining them, is non-trivial. Our method builds on a
recently introduced novel sampling approach that constructs graphs with given
degrees independently (unlike edge-switching Markov Chain Monte Carlo methods)
and efficiently (unlike the configuration model), and extends it to incorporate
the constraint of connectedness. Additionally, we present a simple and elegant
algorithm for directly constructing a single connected realization of a degree
sequence, either as a simple graph or a multigraph. Finally, we demonstrate our
sampling method on a realistic scale-free example, as well as on degree
sequences of connected real-world networks, and show that enforcing
connectedness can significantly alter the properties of sampled networks.
