Ruling invariants for Legendrian graphs

We define ruling invariants for even-valence Legendrian graphs in standard contact three-space. We prove that rulings exist if and only if the DGA of the graph, introduced by the first two authors, has an augmentation. We set up the usual ruling polynomials for various notions of gradedness and prove that if the graph is four-valent, then the ungraded ruling polynomial appears in Kauffman-Vogel's graph version of the Kauffman polynomial. Our ruling invariants are compatible with certain vertex-identifying operations as well as vertical cuts and gluings of front diagrams. We also show that Leverson's definition of a ruling of a Legendrian link in a connected sum of $S^1 \times S^2$'s can be seen as a special case of ours.