Locating an absolute center on graphs that are almost trees

Physical models that exhibit a treelike graph structure frequently arise in telecommunication networks, interstate highway systems, and computer communication networks. Treelike graphs are characterized and properties related to the absolute center problem are analyzed that are preserved under two transformations of the graph. Utilizing these preserved properties, a linear time algorithm is presented for the absolute center problem whenever the underlying graph is homeomorphic to a treelike 3-cactus graph.