Cutwidth of Split Graphs and Threshold Graphs

We give a linear-time algorithm to compute the cutwidth of threshold graphs, thereby resolving the computational complexity of cutwidth on this graph class. Threshold graphs are a well-studied subclass of interval graphs and of split graphs, both of which are unrelated subclasses of chordal graphs. To complement our result, we show that cutwidth is NPcomplete on split graphs, and consequently also on chordal graphs. The cutwidth of interval graphs is still open, and only very few graph classes are known so far on which polynomialtime cutwidth algorithms exist. Thus we contribute to define the border between graph classes on which cutwidth is polynomially solvable and on which it remains NP-complete.