Solving the Euclidean Bottleneck Biconnected Edge Subgraph Problem by 2-Relative Neighborhood Graphs

Chang, MS., C.Y. Tang and R.C.T. Lee, Solving the Euclidean bottleneck biconnected edge subgraph problem by 2-relative neighborhood graphs, Discrete Applied Mathematics 39 (1992) 1-12. In this paper, we discuss the Euclidean bottleneck biconnected edge subgraph problem. We shall first define a k-relative neighborhood graph which is similar to the relative neighborhood graph first proposed by Toussaint. In a k-relative neighborhood graph, a lune contains less than k points. We then show that there exists a solution of the Euclidean bottleneck biconnected edge subgraph problem which is a subgraph of the 2-relative neighborhood graph. With this information , we propose an algorithm to find a Euclidean bottleneck biconnected edge subgraph as follows: (1) Construct a 2-relative neighborhood graph. (2) Use the binary search technique on the sorted edge sequence of the 2-relative neighborhood graph to find a Euc!idean bottleneck biconnected edge subgraph. The construction of the 2-relative neighborhood graph takes 0(n2) time and there are C(n) edges in a 2-relative neighborhood graph. The testing of biconnectivity takes O(n) time since the number of edges is O(n). The total time complexity of our algorithm is 0(n2).