A Characterization of Planar Oriented Graphs

The term “minimal nonplanar subgraph” of a graph G refers to a nonplanar subgraph N of G having the property that each proper subgraph of N is planar. A well-known result of graph theory is that there are two types of minimal nonplanar subgraphs. These types of graphs are usually known as Kuratowski graphs. This paper considers the implications of requiring the edges of the graph to be ordered relative to the nodes of the graph. With this additional condition the minimal nonplanar subgraphs can be characterized as the union of two basis vectors of the graph’s cycle-space.