Graph-aided methods for accommodating the estimation of interactions in factorial experiments have become popular among industrial users. Notable among them is the method of linear graphs due to Taguchi. Previously, some shortcomings of Taguchi's linear graphs were pointed out and an alternative method was proposed. By extending this method some new graphs for three-level fractional factorial designs have been developed. The proposed graphs have two new features. First, each edge of the graph can have one or two lines representing the two components of interaction in a three-level design, and second, there are two types of vertices and lines. A collection of graphs is given for 27-run and 81-run designs.