Recognizing Interval Bigraphs by Forbidden Patterns
Arash Rafiey

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.
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