Characterising the linear clique-width of a class of graphs by forbidden induced subgraphs

We study the linear clique-width of graphs that are obtained from paths by disjoint union and adding true twins. We show that these graphs have linear clique-width at most 4, and we give a complete characterisation of their linear clique-width by forbidden induced subgraphs. As a consequence, we obtain a linear-time algorithm for computing the linear clique-width of the considered graphs. Our results extend the previously known set of forbidden induced subgraphs for graphs of linear clique-width at most 3.