Geodesic
Note on Transformations of Posets with the Same Upper Bound Graph and Minimal Elements
Bulletin of the Malaysian Mathematical Society, Tome 29 (2006) no. 2
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Two posets with the same canonical poset and the same upper bound graph can be transformed into each other by a finite sequence of two kinds of transformations, called x y -additions and x y -deletions on minimal elements.
Classification : 05C62. 
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     author = {Kenjiro Ogawa and Morimasa Tsuchiya},
     title = {Note on {Transformations} of {Posets} with the {Same} {Upper} {Bound} {Graph} and
      {Minimal} {Elements}},
     journal = {Bulletin of the Malaysian Mathematical Society},
     year = {2006},
     volume = {29},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/BMMS_2006_29_2_a7/}
}
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      Minimal Elements
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Kenjiro Ogawa; Morimasa Tsuchiya. Note on Transformations of Posets with the Same Upper Bound Graph and
      Minimal Elements. Bulletin of the Malaysian Mathematical Society, Tome 29 (2006) no. 2. http://geodesic.mathdoc.fr/item/BMMS_2006_29_2_a7/