In this paper, motivated by a question posed by H. de Alba and D. T. Hoang, 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.

中文翻译：

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强双凸图中的诱导匹配和一些代数应用

在本文中，受 H. de Alba 和 D. T. Hoang 提出的问题的启发，我们引入了强双凸图作为弱弦图和二分图的子类。我们给出了一个线性时间算法来寻找此类图的诱导匹配，并且我们证明该算法确实给出了最大诱导匹配。应用该算法，我们提供了一个强双凸图，其（单项式）边缘理想不允许唯一的极值 Betti 数。使用这个构造的图，我们提供了一个无限的所谓闭图（也称为真区间图），其二项式边理想没有唯一的极值 Betti 数。这尤其回答了上述问题。