Determining finite graphs by their large Whitney levels

AbstractLet G be a finite connected graph and C(G) the hyperspace of all subcontinua of G. A Whitney map is a continuous function μ:C(G)→[0,1] such that μ({p})=0 for each p∈G,μ(G)=1 and A⊂B≠A implies that μ(A)<μ(B). A large Whitney level is a set of the form μ−1(t) where 1>t>max{μ(S): S is a proper nonempty connected subgraph of G}. In this paper, we prove the following: Theorem. Let G and H be finite connected graphs with no cut points, then G and H are isomorphic if and only if large Whitney levels for C(G) are homeomorphic to large Whitney levels for C(H)