# Constructing a 4-edge connected graph that is 2-connected but not 3-connected

How can i construct a graph $G$ with these properties:

• $G$ is $4$-edge-connected;
• $G$ is $2$-vertex-connected;
• $G$ is not $3$-vertex-connected.

I have managed to create a number of $4$-edge-connected graphs but they always turn out to be $3$-vertex connected - which I can't have.

essentially, i have used trial and error thus far to construct some 4-edge-connected graphs but i'm struggling to see how i can do this without the resulting graphs also being 3-connected.

4-edge connected meaning a graph in which between any two vertices there are 4 edge-disjoint paths between the two vertices. 2-connected(meaning 2 vertex-connected) means a graph in which between any two vertices there are 2 vertex-disjoint paths.

It's a homework question for a module and could easily be a part of the final year exam - The question explicitly asks for a graph with these three properties, and then to prove that it has these properties.

• 4-edge connected meaning a graph in which between any two vertices there are 4 edge-disjoint paths between the two vertices. 2-connected(meaning 2 vertex-connected) means a graph in which between any two vertices there are 2 vertex-disjoint paths. Feb 26, 2014 at 16:47
• Please add that to the question directly, instead of just being a comment. Feb 26, 2014 at 16:48
• @user3355894 as well adding the comment to the question, could you give some idea of why you think this should exist, and what you have already tried e.g. how you are making these 4 edge connected graphs that turn out to be three vertex connected Feb 26, 2014 at 17:01
• Take two of the 4-edge-connected graph and attache them at two vertices you get a 2-connected graph.
– hbm
Feb 26, 2014 at 17:16

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I hope this helps $\ddot\smile$