Words for the Graphs with Permutation-Representation Number at most Three

TL;DR

This paper explores the permutation-representation number of graphs, providing new algorithms and characterizations for trees, cycles, and book graphs with a focus on those with a number at most three.

Contribution

It introduces a polynomial-time algorithm for representing trees permutationally and determines the permutation-representation number for specific graph classes.

Findings

Polynomial-time algorithm for trees

Representation of even cycles

Permutation-representation number of book graphs

Abstract

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.