Clique-width with an inactive label

An inactive label in a clique-width expression cannot be used to create edges, and vertices that are labelled inactive have already received their incident edges. We study properties of clique-width expressions with inactive labels. The main results are: a characterisation of the distance-hereditary graphs by their clique-width expressions, a characterisation of the linear clique-width of disconnected graphs, and the complete set of disconnected minimal graphs of linear clique-width at least 4.