On powers of interval graphs and their orders

Abstract

It was proved by Raychaudhuri in 1987 that if a graph power $G^{k-1}$ is an interval graph, then so is the next power $G^k$. This result was extended to $m$-trapezoid graphs by Flotow in 1995. We extend the statement for interval graphs by showing that any interval representation of $G^{k-1}$ can be extended to an interval representation of $G^k$ that induces the same left endpoint and right endpoint orders. The same holds for unit interval graphs. We also show that a similar fact does not hold for trapezoid graphs.Comment: 4 pages, 1 figure. It has come to our attention that Theorem 1, the main result of this note, follows from earlier results of [G. Agnarsson, P. Damaschke and M. M. Halldorsson. Powers of geometric intersection graphs and dispersion algorithms. Discrete Applied Mathematics 132(1-3):3-16, 2003]. This version is updated accordingl