For any two points p and q in the Euclidean plane, define LUN, = {ulv E R2,dp, < d, and d,, < dpq}, where d,, is the Euclidean distance between two points u and u. Given a set of points V in the plane, let LUN, ( V ) = V f l LUN, . Toussaint defined the relative neighborhood graph of V, denoted by RNG( V ) or simply RNG, to be the undirected graph with vertices V such that for each pair p,q E V, (p,q) is an edge of
RNG( V ) if and only if LUN, (V ) = 4. The relative neighborhood graph has several applications in pattern recognition that have been studied by Toussaint. We shall generalize the idea of RNG to define the k-relative neighborhood graph of V, denoted by kRNG( V ) or simply kRNG, to be the undirected graph with vertices V such that for each pair p,q E V, (p,q) is an edge of kRNG( V ) if and only if I LUN, ( V)l c k , for some fixed positive number k. It can be shown that the number of edges of a kRNG is less than O(kn). Also, a kRNG can be constructed in time. Let E, = {e,Jp E V and q E V}. Then G, = ( K E , ) is a complete graph. For any subset F of E,, define the maximum distance of F as maxeWEfd,.CCC 0364-9024/91/050543-15$04.00 544 JOURNAL O F GRAPH THEORY G, whose maximum distance is the minimum among all Hamiltonian cycles in graph G, . We shall prove that there exists a Euclidean bottleneck Hamiltonian cycle which is a subgraph of 20RNG( V ) . Hence, 20RNGs are Hamiltonian.
