In this paper, firstly, we reconcile the graphs with permutation-representation number at most two. While the characterization of the class of graphs with permutation-representation number at most three is an open problem, we show that the trees and the even cycles belong to this class. In this connection, we give polynomial-time algorithms for obtaining words representing the trees and the even cycles permutationally.

翻译：本文首先对置换表示数至多为二的图类进行调和。尽管置换表示数至多为三的图类之刻画仍是一个开放问题，我们证明树与偶环均属于该图类。基于此，我们给出多项式时间算法，用以构造表示树与偶环的置换词。