Connectedness, Trees, and Hypergraphs

Abstract

The first section, Sec. 3.1, is the more important one for this chapter. It defines the various ranks of connectedness for transfinite graphs, presents the critical Condition 3.1-2 that insures that connectedness is transitive, and shows how at each rank this partitions a transfinite graph into nonoverlapping subgraphs, called “sections.” Sec. 3.2 explains how a transfinite tree can be easily contracted to and expanded from a conventional tree. Sec. 3.3 relates the v-nodes and (y-1)-sections of a v-graph to a hypergraph and gives an example of how a result from hypergraph theory can be transferred to transfinite graph theory.