Before revision:  We extend this model to the class of mixed unit interval graphs . The original bubble model was used by Boyaci, Ekim, and Shalom for proving the polynomiality of the MaxCut problem on unit interval graphs. However, we found a significant mistake in the proof which seems to be hardly repairable. Moreover, we demonstrate the advantages of such a model by providing a subexponential-time algorithm  solving the MaxCut problem on mixed unit interval graphs using our extended version of the bubble model .
After revision:  We extend this model to the class of mixed unit interval graphs and demonstrate the advantages of such a model by providing a subexponential-time algorithm solving the MaxCut problem on mixed unit interval graphs using our extended version of the bubble model .