Efficient Algorithm for Checking 2-Chordal (Doubly Chordal) Bipartite Graphs

TL;DR

This paper introduces an efficient algorithm to identify 2-chordal bipartite graphs, extends it to k-chordal bipartite graphs, and proves complexity results for these classes.

Contribution

It presents a novel algorithm for detecting 2-chordal bipartite graphs and extends the approach to k-chordal bipartite graphs, establishing their complexity classifications.

Findings

The algorithm efficiently determines 2-chordal bipartite graphs.

No nontrivial k-chordal bipartite graphs exist for k ≥ 4.

Both 2-chordal and 3-chordal bipartite problems are in P.

Abstract

We present an algorithm for determining whether a bipartite graph $G$ is 2-chordal (formerly doubly chordal bipartite). At its core this algorithm is an extension of the existing efficient algorithm for determining whether a graph is chordal bipartite. We then introduce the notion of $k$-chordal bipartite graphs and show by inductive means that a slight modification of our algorithm is sufficient to detect this property. We show that there are no nontrivial $k$-chordal bipartite graphs for $k \geq 4$ and that both the 2-chordal bipartite and 3-chordal bipartite problem are contained within complexity class P.