A characterization of bicyclic but not two-cyclic graphs with a cut vertex as self vertex switching

Abstract

A vertex v V(G) is said to be a self vertex switching of G if G is isomorphic to G^v, where G^v is the graph obtained from G by deleting all edges of G incident to v and adding all edges incident to v which are not in G. A graph G is called a bicyclic but not two-cyclic graph if it is a connected (p, p+1) graph with exactly three cycles. Trees, forests, unicyclic graphs and two-cyclic graphs, with a self vertex switching are characterized in the literature. In this paper, we characterize bicyclic but not two-cyclic graphs with a cut vertex as a self vertex switching. © 2018 Utilitas Mathematica Publishing Inc. All rights reserved.