Extensions of Barrier Sets to Nonzero Roots of the Matching Polynomials

TL;DR

This paper introduces the concept of $ heta$-barrier sets in matching theory, extending the classical barrier sets to nonzero roots of matching polynomials, and establishes related properties and formulas.

Contribution

It defines $ heta$-barrier sets, shows their similarity to barrier sets, and generalizes Berge's Formula for these sets in graphs.

Findings

$ heta$-barrier sets are similar to barrier sets

Generalized Berge's Formula for $ heta$-barrier sets

Characterization of $ heta$-special vertices

Abstract

In matching theory, barrier sets (also known as Tutte sets) have been studied extensively due to its connection to maximum matchings in a graph. In this paper, we first define $\theta$-barrier sets. Our definition of a $\theta$-barrier set is slightly different from that of a barrier set. However we show that $\theta$-barrier sets and barrier sets have similar properties. In particular, we prove a generalized Berge's Formula and give a characterization for the set of all $\theta$-special vertices in a graph.