On powers of circular arc graphs and proper circular arc graphs

In this paper it is shown that the class K of circular are graphs is closed under the operation of power; this means, G is an element of K implies G(k) is an element of K for any positive integer k. Furthermore, G(k) is an element of K implies G(k+2) is an element of K and if diam(G(k)) greater than or equal to 4 then G(k) is an element of K implies G(k+1) is an element of K. Another result on the powers of directed graphs is that if G(k) belongs to the class of in-, out-, or local tournaments, then so does G(k+1). This result is used to prove the closedness of proper circular are graphs under the power operation.