Homeomorphism (graph theory)
In graph theory, two graphs and are homeomorphic if there is a graph isomorphism from some subdivision of to some [|subdivision] of. If the edges of a graph are thought of as lines drawn from one vertex to another, then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if their diagrams are homeomorphic in the topological sense.
Subdivision and smoothing
In general, a subdivision of a graph G is a graph resulting from the subdivision of edges in G. The subdivision of some edge e with endpoints yields a graph containing one new vertex w, and with an edge set replacing e by two new edges, and. For directed edges, this operation shall preserve their propagating direction.For example, the edge e, with endpoints :
[Image:Graph subdivision step1.svg|150px|class=skin-invert]
can be subdivided into two edges, e1 and e2, connecting to a new vertex w of degree-2, or indegree-1 and outdegree-1 for the directed edge:
[Image:Graph subdivision step2.svg|150px|class=skin-invert]
Determining whether for graphs G and H, H is homeomorphic to a subgraph of G, is an NP-complete problem.
Reversion
The reverse operation, smoothing out or smoothing a vertex w with regards to the pair of edges incident on w, removes both edges containing w and replaces with a new edge that connects the other endpoints of the pair. Here, it is emphasized that only degree-2 vertices can be smoothed. The limit of this operation is realized by the graph that has no more degree-2 vertices.For example, the simple connected graph with two edges, e1 and e2 :
[Image:Graph subdivision step2.svg|150px|class=skin-invert]
has a vertex that can be smoothed away, resulting in:
[Image:Graph subdivision step1.svg|150px|class=skin-invert]
Barycentric subdivisions
The barycentric subdivision subdivides each edge of the graph. This is a special subdivision, as it always results in a bipartite graph. This procedure can be repeated, so that the nth barycentric subdivision is the barycentric subdivision of the n−1st barycentric subdivision of the graph. The second such subdivision is always a simple graph.Embedding on a surface
It is evident that subdividing a graph preserves planarity. Kuratowski's theorem states thatIn fact, a graph homeomorphic to K5 or K3,3 is called a Kuratowski subgraph.
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. For example, consists of the Kuratowski subgraphs.
Example
In the following example, graph G and graph H are homeomorphic.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