  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.
