Ruling invariants for Legendrian graphs

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