I need to Show that Every edge of a 2- Connected graph is either contractible or Deletable.
Contractible Edge:- To contract an Edge, remove it and its end points(vertices) should be merged along with other edges incident on them.
Deletable Edges:- A edge e is Deletable from Graph G if we remove that edge from Graph it don't change the Connectivity(Kappa) of that graph. i.e. Kappa(G)=Kappa(G-e)
An Example of Deletable and Contractible Edges can be found in Bondy and Murty.
I am starting as this..
Since given that the graph is two connected, there exists atleast two distinct internally disjoint paths between any pair of vertices. There fore Kappa Prime which is edge connectivity is greater than or equal to 2 . Thus G don't have cut edges.
So, an edge can be a loop or a Parallel Edge or a normal edge. I am able to see that a loop is either contractible or deletable.
So, I am not sure on how to proceed from here. Can someone kindly guide me.
Thanks,