If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in diagrams), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if their diagrams are homeomorphic in the topological sense.
Determining whether for graphs G and H, H is homeomorphic to a subgraph of G, is an NP-complete problem.
The limit of this operation is realized by the graph that has no more degree-2 vertices.
The barycentric subdivision subdivides each edge of the graph.
This is a special subdivision, as it always results in a bipartite graph.
It is evident that subdividing a graph preserves planarity.
A generalization, following from the Robertson–Seymour theorem, asserts that for each integer g, there is a finite obstruction set of graphs
such that a graph H is embeddable on a surface of genus g if and only if H contains no homeomorphic copy of any of the
If G′ is the graph created by subdivision of the outer edges of G and H′ is the graph created by subdivision of the inner edge of H, then G′ and H′ have a similar graph drawing: Therefore, there exists an isomorphism between G' and H', meaning G and H are homeomorphic.
The directed edges are shown to have an intermediate arrow head.