The excluded minors for isometric realizability in the plane

Let G be a graph and p ∈ [1,∞]. The parameter fp(G) is the least integer k such that for all m and all vectors (rv)v∈V (G) ⊆ R , there exist vectors (qv)v∈V (G) ⊆ R k satisfying ‖rv − rw‖p = ‖qv − qw‖p, for all vw ∈ E(G). It is easy to check that fp(G) is always finite and that it is minor monotone. By the graph minor theorem of Robertson and Seymour [9], there are a finite number of excluded minors for the property fp(G) 6 k. In this paper, we determine the complete set of excluded minors for f∞(G) 6 2. The two excluded minors are the wheel on 5 vertices and the graph obtained by gluing two copies of K4 along an edge and then deleting that edge. Due to an isometry between the corresponding metric spaces, this also characterizes the graphs G with f1(G) 6 2.