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

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.