The graphs with permutation-representation number (\textit{prn}) at most two are known. While a characterization for the class of graphs with the \textit{prn} at most three is an open problem, we summarize the graphs of this class that are known so far. Although it is known that the \textit{prn} of trees is at most three, in this work, we devise a polynomial-time algorithm for obtaining a word representing a given tree permutationally. Consequently, we determine the words representing even cycles. Contributing to the class of graphs with the \textit{prn} at most three, we determine the \textit{prn} as well as the representation number of book graphs.

翻译：已知置换表示数（prn）不超过2的图。尽管prn不超过3的图类的刻画仍是一个开放问题，我们归纳了目前已知的属于此类图的所有图。虽然已知树的prn不超过3，但在本文中，我们设计了一个多项式时间算法，用于获得表示给定树的置换词语。由此，我们确定了表示偶环的词语。作为对prn不超过3的图类的贡献，我们确定了书图（book graphs）的prn及其表示数。