Facets for the single node fixed-charge network set with a node set-up variable

We consider a variant of the well-known Single Node Fixed-Charge Network
(SNFCN) set where a set-up variable is associated with the node, indicating
whether the node is open or not. This set arises as a relaxation of several practical
mixed integer problems. We relate the polyhedral structure of this variant with the
polyhedral structure of the SNFCN set. We show that in the presence of the node setup variable new facet-defining inequalities appear and establish the relation between the new family of inequalities with the flow cover inequalities. For the constant capacitated case we provide a full polyhedral description of the convex hull of the given set.