Interpolation of point configurations in the discrete plane

Defining distances over finite fields formally by ||x-y||: = (x₁-y₁) ²+ + (xd-yd) ² for x, y Fqᵈ, distance problems naturally arise in analogy to those studied by Erdos and Falconer in Euclidean space. Given a graph G and a set E Fq², let G (E) be the generalized distance set corresponding to G. In the case when G is the complete graph on k+1 vertices, Bennett, Hart, Iosevich, Pakianathan, and Rudnev showed that when |E| q^d-d-1{k+1}, it follows that |G (E) | cq^k+1{2}. In the case when k=d=2, the threshold can be improved to |E| q^8{5}. Moreover, Jardine, Iosevich, and McDonald showed that in the case when G is a tree with k+1 vertices, then whenever E Fqᵈ, d 2 satisfies |E| Cₖq^d+1{2}, it follows that G (E) =Fqᵏ. In this paper, we present a technique which enables us to study certain graphs with both rigid and non-rigid components. In particular, we show that for E Fq², q=pⁿ, n odd, p 3 \ mod \ 4, and G is the graph consisting of two triangles joined at a vertex, then whenever |E| q^12{7}, it follows that |G (E) | cq⁶.