Graphs of linear clique-width at most 3

A graph has linear clique-width at most k if it has a clique-width expression using at most k labels such that every disjoint union operation has an operand which is a single vertex graph. We give the first characterisation of graphs of linear clique-width at most 3, and we give the first polynomial-time recognition algorithm for graphs of linear clique-width at most 3. In addition, we present new characterisations of graphs of linear clique-width at most 2. We also give a layout characterisation of graphs of bounded linear clique-width; a similar characterisation was independently shown by Gurski and by Lozin and Rautenbach.