Recognizing Interval Bigraphs by Forbidden Patterns

TL;DR

This paper presents an efficient O(nm) algorithm for recognizing interval bigraphs, improving upon previous methods by leveraging an ordering characterization and pair-digraph structure.

Contribution

It introduces a novel O(nm) time algorithm for recognizing interval bigraphs based on forbidden pattern characterization and pair-digraph analysis.

Findings

The algorithm runs in O(nm) time, significantly faster than previous methods.

It effectively identifies interval bigraphs using pair-digraph decomposition.

The approach clarifies difficult cases and isolates them for efficiency.

Abstract

Let H be a connected bipartite graph with n nodes and m edges. We give an O(nm) time algorithm to decide whether H is an interval bigraph. The best known algorithm has time complexity O(nm^6(m + n) \log n) and it was developed in 1997 [18]. Our approach is based on an ordering characterization of interval bigraphs introduced by Hell and Huang [13]. We transform the problem of finding the desired ordering to choosing strong components of a pair-digraph without creating conflicts. We make use of the structure of the pair-digraph as well as decomposition of bigraph H based on the special components of the pair-digraph. This way we make explicit what the difficult cases are and gain efficiency by isolating such situations.