How to cluster a network?

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Hello guys,
If I have a network ( Lets call it x) and it is described by the following edges:
x=[1 2 3 4 5 6]; %vertices in the network
Edg =[1 2; 1 3 ; 2 3 ; 3 4 ; 4 5;6 3; 5 6]; %edges
A=graph(Edg(:,1),Edg(:,2));
plot(A)
The graph A looks like this :
If x is devided into two areas (we know the nodes in each area):
x1=[1 2 3];
x2=[3 4 5 6]; %the common vertex is 3
%edges is the same
Now i want to plot each area, so the graph above will be splitted into two graphs and vertix 3 is in both graphs.
To do this i need to write a code which gives the edges in each area, like this:
Edges for x1 [ 1 2; 2 3; 1 3]. Edges for x2 [ 3 6; 3 4; 4 5 ; 5 6].

Akira Agata on 14 Jul 2019
How about applying biconncomp function?
The following is an example:
% Sample Graph
s = [1 1 2 2 3 4 4 5 6 6 7 7 8];
t = [2 3 3 4 4 5 7 6 7 10 8 9 9];
G = graph(s,t);
% Visualize G
figure
plot(G);
% Apply biconncomp function
bincell = biconncomp(G, 'OutputForm', 'cell');
n = length(bincell);
% Visualize the separated graphs
figure
for ii = 1:n
subplot(2,2,ii)
plot(subgraph(G, bincell{ii}), 'NodeLabel', bincell{ii});
end
Original Graph:
Separated graphs:
abdullah nasser abdullah on 17 Jul 2019
Many thanks for your answer Mr. Akira,
Your answer perfectly fit if there is only one coupled node between any two areas,
but lets consider that i have more than coupled node like the following:
x=[1 2 3 4 5 6]; %vertices in the network
Edg =[1 2; 1 4 ; 2 3 ; 3 4 ; 4 5;6 3; 5 6]; %edges
G=graph(Edg(:,1),Edg(:,2));
plot(G)
And we want the paritioning to be like this:
Where 3 and 4 are the common node between the two areas.
I want something that work for larger systems; where i know the nodes for each area and i know the coupled nodes between them as well.