  In this paper, motivated by a question posed in \cite{AH}, we introduce
strongly biconvex graphs as a subclass of weakly chordal and bipartite graphs.
We give a linear time algorithm to find an induced matching for such graphs and
we prove that this algorithm indeed gives a maximum induced matching. Applying
this algorithm, we provide a strongly biconvex graph whose (monomial) edge
ideal does not admit a unique extremal Betti number. Using this constructed
graph, we provide an infinite family of the so-called closed graphs (also known
as proper interval graphs) whose binomial edge ideals do not have a unique
extremal Betti number. This, in particular, answers the aforementioned question
in \cite{AH}.
