Forbidden subgraphs of graphs uniquely Hamiltonian-connected from a vertex

A graph G is called uniquely Hamiltonian-connected from a vertex upsilon if, for every vertex u not equal upsilon, there is exactly one upsilon - u Hamiltonian path in G. We show that if G is uniquely Hamiltonian-connected from upsilon and H is a subgraph of G - upsilon then \E(H)\ less than or equal to (3\V(H)\ - 2)/2. The bound on the number of edges is best possible in that there exists graphs H with exactly [(3\V(H)\ - 2)/2] edges which are forbidden and others which can occur as subgraphs. (C) 1998 Elsevier Science B.V. All rights reserved.