Recognizing Powers of Proper Interval, Split, and Chordal Graphs

In this paper, we study the complexity of recognizing powers of chordal graphs and its subclasses. We present the first polynomial time algorithm to recognize squares of proper interval graphs and give an outline of an algorithm to recognize kth powers of proper interval graphs for every natural number k. These are the first results of this type for a family of graphs that contains arbitrarily large cliques. On the other hand, we show the NP-completeness of recognizing squares of chordal graphs, recognizing squares of split graphs, and recognizing chordal graphs that are squares of some graph.