Exercise 7.4.4:

Edge connectivity between two vertices. About Two vertices v and w in a graph G are said to be 2-edge-connected if the removal of any edge in the graph leaves v and w in the same connected component.

(a) Prove that G is 2-edge-connected if every pair of vertices in G are 2-edge-connected.

(b) Is being 2-edge-connected a transitive property? That is, if v and w are 2-edge-connected and w and y are 2 edge-connected, can you conclude that v and y are 2-edge-connected? Prove your answer. You can use the fact that if there is a walk from vertex v to vertex w, then there is a path from vertex v to vertex w. (

c) Two vertices v and w in a graph G are said to be 2-vertex-connected if the removal of any vertex in the graph besides v and w leaves v and w in the same connected component. Is being 2-vertex-connected a transitive property? That is, if v and w are 2-vertex-connected and w and y are 2-vertex-connected, can you conclude that v and y are 2-vertex-connected? Justify your answer. You can use the fact that if there is a walk from vertex v to vertex w, then there is a path from vertex v to vertex w.