The 1-Center Problem: Exploiting Block Structure

A block of a graph is a maximal nonseparable subgraph. We show how the knowledge of block structure can be used to help solve the nonlinear 1-center problem on graphs which are more general than trees. We give an efficient algorithm which either finds a unique 1-center at some vertex, or else localizes the search for all 1-centers to a single block. The algorithm makes use of an associated graph, called a blocking graph, which is a tree, and iteratively orients arcs in the blocking graph to “point the way” to a block which contains all 1-centers.