Abstract
Let G be a graph and p is an element of[1,infinity]. The parameter fp(G) is the least integer k such that for all m and all vectors (r(v))(v is an element of V(G)) subset of R-m, there exist vectors (q(v))(v is an element of V(G))subset of R-k satisfying parallel to r(v)-r(w)parallel to p=parallel to q(v)-q(w)parallel to(p), for all vw is an element of E(G). It is easy to check that f(p)(G) is always finite and that it is minor monotone. By the graph minor theorem of Robertson and Seymour, there are a finite number of excluded minors for the property f(p)(G)<= k. In this paper, we determine the complete set of excluded minors for f(infinity)(G)<= 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. We also show that the same two graphs are the complete set of excluded minors for f(1)(G)<= 2. In addition, we give a family of examples that show that f(infinity) is unbounded on the class of planar graphs and f(infinity) is not bounded as a function of tree-width.