drawing a cubic graph I am required to draw a cubic graph for which k(G) =1 and k'(G)=2. My attempts give me such a graph  have resulted with k(G)=1 and k'(G) also equal to 1. Whenever attempt is made to get k'(G)=2, the graph does not disconnect by removing 1 vertex. Theoretically, the question satisfies the criteria  
k(G)$\leq$k'(G)$\leq$$\delta$(G)
Since I am unable to draw such a graph, I had a doubt if it is possible to draw such a graph at all even though above condition is being satisfied.
(i)   Is it possible to draw the said graph? If the answer is yes, thats good enough.. I will keep trying and find the solution.
(ii) Are there any conditions when even though  k(G)$\leq$k'(G)$\leq$$\delta$(G) is satisfied, it is not possible to draw a graph?
Request guide
 A: For a cubic/3-valent/3-regular graph, edge connectivity is the same as vertex connectivity.
Proving if G is a 3-regular graph, then the size of edge cut equals size of min size of vertex cut
The idea is that, given a collection of vertices that cut the graph into two components, each vertex $v$ will have one edge $e_v$ to one component and two edges to the other component.  Removing the collection of $e_v$ separates the graph.  This argument implies $\kappa'(G)\leq \kappa(G)$, and you already seem to have $\kappa(G)\leq\kappa'(G)$.

A possibly useful observation for understanding connectivity or searching for graphs with particular connectivity: If you have a connected cubic graph that is 2-connected, then you can cut those edges in half, and join the edges up within each component to get two graphs that are at least as connected.  Conversely, if you have two cubic graphs, you can slice an edge in each of them and perform a connect sum to get a 2-connected graph.
Similarly, if you have three edges which disconnect the graph, then you can slice those edges in half, connect the edges to two newly created vertices and get two graphs that are at least as connected.  (Slicing off a single vertex, one of the graphs you get is a theta graph.)  The inverse operation is a connect sum at a vertex: slice off a vertex of each graph, and join the three edges in one graph with the three in the other.
Define a prime graph to be a connected cubic graph that is 3-connected so that this 3-slicing operation only gives the graph and a theta graph back.  Every connected cubic graph can be obtained by some sequence of edge and vertex connect sums.
