A Note on Near-factor-critical Graphs

TL;DR

This paper characterizes near-factor-critical graphs, showing that connected near-factor-critical graphs are exactly those with perfect matchings, and provides a characterization for disconnected cases.

Contribution

It establishes a precise characterization of near-factor-critical graphs, linking this property to the existence of perfect matchings in connected graphs.

Findings

Connected near-factor-critical graphs are exactly those with perfect matchings.

Disconnected near-factor-critical graphs are characterized within the paper.

Provides a complete characterization of near-factor-critical graphs.

Abstract

A near-factor of a finite simple graph $G$ is a matching that saturates all vertices except one. A graph $G$ is said to be near-factor-critical if the deletion of any vertex from $G$ results in a subgraph that has a near-factor. We prove that a connected graph $G$ is near-factor-critical if and only if it has a perfect matching. We also characterize disconnected near-factor-critical graphs.