Rectangular Diagrams of Legendrian Graphs

In this paper Legendrian graphs in $(\mathbb{R}^3,\xi_{\mathrm{st}})$ are considered modulo Legendrian isotopy and edge contraction. To a Legendrian graph we associate a (generalized) rectangular diagram --- a purely combinatorial object. Moves of rectangular diagrams are introduced so that equivalence classes of Legendrian graphs and rectangular diagrams coincide. Using this result we prove that the classes of Legendrian graphs are in one-to-one correspondence with fence diagrams modulo fence moves introduced by Rudolph.