Eulerian Location Problems

The distance from a vertex u to a vertex v in a connected graph G is the length of a shortest u-v path in G. The distance of a vertex v of G is the sum of the distances from v to the vertices of G. For a vertex v in a 2-edge-connected graph G, we define the edge-deleted distance of v as the maximum distance of v in G -e over all edges e of G. A vertex is an edge-deleted distance stable vertex if the difference between its edge-deleted distance and distance is 1. A 2-edge-connected graph G is an edge-deleted distance stable graph if each vertex of G is an edge-deleted distance stable vertex. In this paper, we investigate the edge-deleted distance of vertices and describe properties of edge-deleted distance stable graphs. © 1999 John Wiley & Sons, Inc. Networks 34: 279-282, 1999