Complete characterization of bicyclic minimal configurations

Abstract

Sciriha first proposed the notion of minimal configurations. A graph G with nullity one is called a minimal configuration if no two vertices in the periphery are adjacent and deletion of any vertex in the periphery increases the nullity. Recently, minimal configurations in special class of graphs, such as trees and unicyclic graphs, have been studied. The set of bicyclic graphs, denoted by , can be partitioned into two subsets: the set ℬ* of graphs which contain induced ∞-graphs, and the set ℬ** of graphs which contain induced θ-graphs. Nath described the minimal configurations in ℬ*. In this article, we will describe the minimal configurations in the set ℬ**. These results together give a complete characterization of the minimal configurations in .

Keywords:

• singular graphs
• minimal configuration
• bicyclic graph

AMS Subject Classfication:

• 05C50

Acknowledgements

The authors are grateful to the referees for careful reading and many valuable suggestions, which greatly improved the presentation of the article. The first author sincerely thanks Professor Milan Nath for sending us their paper 4. Research was supported by NNFS of China (No. 11071088).