Algorithm for generating a tree decomposition

Question

I want to construct a tree decomposition: http://en.wikipedia.org/wiki/Tree_decomposition and I have the chordal graph and a perfect elimination ordering. I am following adv

Answer 1

So that was my (bad, as it turns out) advice. Your tree decomposition has some cliques that aren't maximal, i.e., {2, 0, 1}, {0, 1}, and {1}. The list of candidate cliques is

{4, 5, 6} (maximal)
{3, 2} (maximal)
{5, 6, 2, 0} (maximal)
{7, 2, 1} (maximal)
{6, 2, 0, 1} (maximal)
{2, 0, 1} (not maximal: subset of {6, 2, 0, 1})
{0, 1} (not maximal: subset of {6, 2, 0, 1})
{1} (not maximal: subset of {6, 2, 0, 1})

Then the decomposition should look like

                {3, 2}
                  |
{4, 5, 6}----{5, 6, 2, 0}
                  |
             {7, 2, 1}
                  |
             {6, 2, 0, 1},

which is wrong as well, since the 0 vertices aren't connected. Sorry about that.

What I should have done is to set aside the maximality condition for the moment and to connect each clique K to the next candidate seeded with a vertex belonging to K. This ensures that every vertex in K that exists in at least one other subsequent clique also appears in the target of the connection. Then the decomposition looks like this

{4, 5, 6}----{5, 6, 2, 0}
                  |
             {6, 2, 0, 1}
                  |
   {3, 2}----{2, 0, 1}----{7, 2, 1}
                  |
                {0, 1}
                  |
                {1}

and you can splice out the non-maximal cliques by checking, for each clique in reverse order, whether it's a superset of its parent, and if so, reparenting its parent's children to it.

{4, 5, 6}----{5, 6, 2, 0}
                  |
   {3, 2}----{6, 2, 0, 1}----{7, 2, 1}