Clique-width of full bubble model graphs

Bubble models are 2-dimensional representations of proper interval graphs. We consider proper interval graphs that have bubble models of specific properties. We characterise the maximal such proper interval graphs of bounded clique-width and of bounded linear clique-width and the minimal such proper interval graphs whose clique-width and linear clique-width exceed the bounds. As a consequence, we can efficiently compute the clique-width and linear clique-width of the considered graphs.