Induced matchings in strongly biconvex graphs and some algebraic applications
Abstract
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 socalled 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}.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 arXiv:
 arXiv:1905.02640
 Bibcode:
 2019arXiv190502640S
 Keywords:

 Mathematics  Commutative Algebra;
 Mathematics  Combinatorics;
 Primary 05E40;
 13D02;
 Secondary 05C70
 EPrint:
 19 pages, 2 figures