# Induced matchings in strongly biconvex graphs and some algebraic applications

## TL;DR

This paper introduces strongly biconvex graphs, provides a linear time algorithm for maximum induced matchings, and explores algebraic properties of associated edge ideals, answering an open question.

## Contribution

It defines strongly biconvex graphs, develops an efficient algorithm for induced matchings, and investigates algebraic invariants of related edge ideals, addressing a previously open problem.

## Key findings

- Linear time algorithm for maximum induced matching in strongly biconvex graphs
- Existence of strongly biconvex graphs with non-unique extremal Betti numbers
- Infinite family of closed graphs with non-unique extremal Betti numbers

## 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 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}.