  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.
