Cutwidth Bounds via Vertex Partitions

We study the cutwidth measure on graphs and ways to bound the cutwidth of a graph by partitioning its vertices. We consider bounds expressed as a function of two quantities: on the one hand, the maximal cutwidth x of the subgraphs induced by the classes of the partition, and on the other hand, the cutwidth y of the quotient multigraph obtained by merging each class to a single vertex. We consider in particular decomposition of directed graphs into strongly connected components (SCCs): in this case, x is the maximal cutwidth of an SCC, and y is the cutwidth of the directed acyclic condensation multigraph. We show that the cutwidth of a graph is always in O(x + y), specifically it can be upper bounded by 1.5x + y. We also show a lower bound justifying that the constant 1.5 cannot be improved in general