I am studying Clustering algorithm and having problem in "transforming a Bayesian Network into a polytree by removing multiple paths between the two nodes along which evidence may travel" (I am reading "Bayesian Artificial Intelligence" by Kevin B. Korb and Ann E. NIcholson, page 66 - ).

My example network has 8 nodes and their connections are as follows:

(1)->(3)
(1)->(4)
(2)->(4)
(2)->(5)
(2)->(6)
(3)->(5)
(3)->(6)
(3)->(7)
(4)->(5)
(4)->(6)
(4)->(7)
(6)->(8)

Here, I can't find a way to create a "polytree" that has at most one path between any pair of nodes. Main problem is with node#2 to node#4 because the parent (node#2) and child (node#4) share common children (node#5 and node#6). Is it possible to combine a parent and a child node together in the polytree transformation?

To make the question simpler, if I have a 3 node BN with connections (1)->(2), (1)->(3), and (2)->(3), how should I turn it into a polytree?

Please help me. Thank you in advance.