An Upper Bound for the Permutation-representation Number of Bipartite Graphs

A word-representable graph G=(V,E) is a simple graph which can be represented by a word w over its vertex set V such that any two vertices a and b of G are adjacent if and only if a and b alternate in the word w. A word-representable graph can be represented by a word in which each letter appears exactly k times, for some k; the smallest such k is called the representation number of the word-representable graph. If a graph is represented by a word of the form p1p2…pk where each pi is a permutation of its vertices, then the graph is called permutationally representable, and the smallest such k is called the permutation-representation number (in short, prn) of the permutationally representable graph. In the literature, the representation number and the prn were established for certain subclasses of bipartite graphs. In this work, focusing on the prn, we observe the relation between the prn of a comparability graph and the dimension of its induced poset, and note that they have the same value. Accordingly, we ascertain that determining the prn of a bipartite graph is NP-hard. Next, we review existing upper bounds on the prn of comparability graphs and bipartite graphs, and subsequently examine the prn of bipartite graphs using specific constructions we refer to as neighborhood inclusion graphs. Here, we both devise a polynomial-time procedure to construct a word that represents a given bipartite graph permutationally, and also provide an improved upper bound for the prn of bipartite graphs.