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					I.    Different situations when performing Block-Computation-Algorithm. ................ 2
      1. path_node is the only one node on RBC ........................................................ 2
      2. path_node is the first full node on RBC......................................................... 2
      3. one of path_node’s neighbor is a full node .................................................... 2
      4. both neighbors of path_node are full nodes ................................................... 2
      (i.e. path_node is a gap between two contiguous full-nodes) ................................ 2
II.   Different situations when performing Computation of RBC. ................................ 3
      1. path_node is partial node on RBC with opp_bend is also partial. ................. 3
      2. path_node is full node on RBC with opp_bend is partial. ............................. 3
      3. path_node is partial node on RBC with opp_bend is full. ............................. 4
      4. path_node is full node on RBC with opp_bend is also full. .......................... 4
III.   Different situations when drawing out non-planar K33/K5 subgraphs. .............. 5
     1. The returned-value of pt_test in code is -1, -2 or -3. ..................................... 5
     2. The returned-value of pt_test in code is -4. ................................................... 5
      3.    The returned-value of pt_test in code is -5. ................................................... 6
      4.    The returned value of pt_test in code is -6. .................................................... 7
I.   Different situations when performing Block-Computation-Algorithm.
     (path_node refers to currently processed node when traversing to a RBC)
     1. path_node is the only one node on RBC




                                F
                              path_node


           Entire RBC path_node belongs to is embedded into new RBC
     2.    path_node is the first full node on RBC




                                                               na_full_end
                                                         nbr          nbr

                       F                                         F
                     path_node                                  path_node

     3.    one of path_node’s neighbor is a full node



                          F                                           F


                                      na_full_end                           na_full_end
                                                        nbr
                          F                                         F
                    F                                           F
                  path_node                                    path_node

     4.    both neighbors of path_node are full nodes
           (i.e. path_node is a gap between two contiguous full-nodes)


                                                               nbr          nbr
                                                                F           F
                 F               F


 na_full_end                         na_full_end
                                                                                  na_full_end
                                                                F           F
                 F               F                                     F
                         F
                                                                     path_node
                     path_node
          When case 3 & case 4, check if the two ending node of contiguous full
           nodes are the end nodes of RBC. If yes, the whole RBC is embedded into
           new RBC.
II.   Different situations when performing Computation of RBC.
      (path_node refers to currently processed node when traversing to a RBC)
      1. path_node is partial node on RBC with opp_bend is also partial.




                                                       opp_bend
                                    F              P
                                    RBC_head

                     na_full_end
                                            nbr
                                                 P
                                          P          link_nbr
                                       path_node(link_node)

                                        P last_partial




         Cut RBC from path_node to RBC_head and then traverse up from
      RBC_head.
         (last_partialpath_nodelink_nbr(nbr of lath_node) …opp_bend)


      2.   path_node is full node on RBC with opp_bend is partial.




                                                       opp_bend
                                    F              P
                                    RBC_head
                     na_full_end                        P
                                            nbr             link_nbr
                                                   P
                                          F            link_node
                                       path_node

                                          P last_partial




         Cut RBC from nbr of path_node to RBC_head and then traverse up from
      RBC_head.
         (last_partiallink_nbr(nbr of lath_node) …opp_bend)
3.   path_node is partial node on RBC with opp_bend is full.



                                                  opp_bend
                                                   F    na_full_end
                                    F
                                                        P
                                    RBC_head
                 na_full_end
                                                           P
                                              nbr              link_nbr
                                                       P
                                           P        link_node
                                        path_node(link_node)

                                         P last_partial




     Cut RBC from path_node to na_full_end[opp_bend] and then traverse up
     from RBC_head.
     (last_partialpath_nodelink_nbr(nbr of lath_node)
     …na_full_end[opp_bend])


4.   path_node is full node on RBC with opp_bend is also full.



                                      opp_bend
                                        F    na_full_end
                               F
                                             P
                               RBC_head
           na_full_end
                                                   P
                                       nbr             link_nbr
                                              P
                                     F            link_node
                                  path_node

                                     P last_partial




     Cut RBC from nbr of path_node to na_full_end[opp_bend] and then
     traverse up from RBC_head.
     (last_partiallink_nbr(nbr of lath_node) …na_full_end[opp_bend])
III. Different situations when drawing out non-planar K33/K5 sub-graph.
     1. The returned-value of pt_test in code is -1, -2 or -3.
          (Case 1.1 & 1.2 in 「PLANAR implementation- Handbook」)
        An RBCi (RBC with head node i) is encountered when traversing back
    edges during j iteration. After traversing all back edges, we found the two ends of
    RBCi are not marked as full. Thus violate the property (e) of 「PLANAR
    implementation- Handbook」).

                  m

                                                     K3,3
                  k

                                             U        i       k

                  j
                                             S        j       T
                  i
         S                T


                                 partial/empty
             U                      nodes




             full nodes


    2.   The returned-value of pt_test in code is -4.
         A. When connection RBC at i iteration, two terminal nodes (S and T)
              have been identified when traversing a back edge (i-S back edge).
              After traversing another back edge (i-T backedge), we found another
              node T should also identified as terminal node. Thus violate the
              property (c) of 「PLANAR implementation- Handbook」).



                                                       K3,3
                      k

                                                 P        i       k
                      w

                                                 S        U       T
                      i

                      U

                                            terminal nodes
                      P

                                T
     S
     B.   When connection RBC at i iteration, an RBCp (RBC with head node p)
          is traversed and part of it (S to T alone with its boundary) has been
          merged into new RBC with two new ends, S and U. After traversing
          another back edge (i-T backedge), we found another node T in
          previous RBCp should also identified as terminal node. Thus violate
          the property (c) of 「PLANAR implementation- Handbook」).

            m

                                              K3,3
            k

            i                           P       i       k

           U
                                        S       U       T
            P
     S               T


                            terminal nodes




3.   The returned-value of pt_test in code is -5.
     (Case 2.1 in 「PLANAR implementation- Handbook」)
      When connection RBC at j iteration, an RBCi (RBC with head node i) is
traversed and part of it (S to T alone with its boundary) have been merged into
new RBC with two new end nodes, S and i. After traversing another back edge
(j-T backedge), we found another node T in previous RBCp should also identified
as terminal node. Thus violate the property (c) of 「PLANAR implementation-
Handbook」).

             m


             k



             j

             i
     S               T


                             terminal nodes
4.   The returned value of pt_test in code is -6.
     When connection RBC at j iteration, an RBCi (RBC with head node i) is
     traversed and part of it (S to i alone with its boundary) has been merged into
     new RBC with two new ends, S and i. After traversing another back edge
     (j-T backedge), we found another node T in previous RBCi should also
     identified as terminal node. Thus violate the property (c) of 「PLANAR
     implementation- Handbook」).

                n

                                                  K3,3
                m

                 k                          m       j      T

                j
                                            S      i       k
                i
         S               T


                                terminal nodes




                n

                                                  K3,3
                m

                 k                          m          j    i
                j                                           k

                                            S      k        T
                i
         S               T


                                terminal nodes




                n

                                                  K3,3
                m

                 k                          m       j      S

                j
                                            i      k       T
                i
         S               T


                                terminal nodes

				
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posted:11/2/2011
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