Graphs of small bounded linear clique-width ∗

In this paper we give the first characterisation of graphs with linear clique-width at most 3, and we give a polynomial-time recognition algorithm for such graphs. In addition, we give a new characterisation of graphs with linear clique-width at most 2 and a new layout characterisation of linear clique-width in general. Among our results is also a decomposition scheme that preserves the linear clique-width of the decomposable subgraph.