On the Permutation-Representation Number of Bipartite Graphs using Neighborhood Graphs

TL;DR

This paper investigates the permutation-representation number of bipartite graphs, introduces neighborhood graphs as a tool, and provides new bounds and polynomial-time procedures, supporting the conjecture that crown graphs have the highest representation number.

Contribution

It establishes a connection between prn and neighborhood graphs, offers a polynomial-time construction method, and improves upper bounds for bipartite graphs' prn.

Findings

Neighborhood graphs help analyze bipartite graph representations.

A polynomial-time procedure for constructing permutational representations.

Extended crown graphs are studied with new insights into their prn.

Abstract

The problems of determining the permutation-representation number (prn) and the representation number of bipartite graphs are open in the literature. Moreover, the decision problem corresponding to the determination of the prn of a bipartite graph is NP-complete. However, these numbers were established for certain subclasses of bipartite graphs, e.g., for crown graphs. Further, it was conjectured that the crown graphs have the highest representation number among the bipartite graphs. In this work, first, we reconcile the relation between the prn of a comparability graph and the dimension of its induced poset and review the upper bounds on the prn of bipartite graphs. Then, we study the prn of bipartite graphs using the notion called neighborhood graphs. This approach substantiates the aforesaid conjecture and gives us theoretical evidence. In this connection, we devise a polynomial-time procedure to construct a word that represents a given bipartite graph permutationally. Accordingly, we provide a better upper bound for the prn of bipartite graphs. Further, we construct a class of bipartite graphs, viz., extended crown graphs, defined over posets and investigate its prn using the neighborhood graphs.