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 (as they are usually depicted in illustrations), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if they are homeomorphic in the topological sense.[1]
In general, a subdivision of a graph G (sometimes known as an expansion[2]) is a graph resulting from the subdivision of edges in G. The subdivision of some edge e with endpoints {u,v } yields a graph containing one new vertex w, and with an edge set replacing e by two new edges, {u,w } and {w,v }.
For example, the edge e, with endpoints {u,v }:
can be subdivided into two edges, e1 and e2, connecting to a new vertex w of degree-2:
The reverse operation, smoothing out or smoothing a vertex w with regards to the pair of edges (e1, e2) incident on w, removes both edges containing w and replaces (e1, e2) with a new edge that connects the other endpoints of the pair. Here, it is emphasized that only degree-2 (i.e., 2-valent) vertices can be smoothed.
For example, the simple connected graph with two edges, e1 {u,w } and e2 {w,v }:
has a vertex (namely w) that can be smoothed away, resulting in:
Determining whether for graphs G and H, H is homeomorphic to a subgraph of G, is an NP-complete problem.[3]
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.
It is evident that subdividing a graph preserves planarity. Kuratowski's theorem states that
In 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.
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 G' and H', meaning G and H are homeomorphic.
The name arises because and are homeomorphic as graphs if and only if they are homeomorphic as topological spaces
Definition 20. If some new vertices of degree 2 are added to some of the edges of a graph G, the resulting graph H is called an expansion of G.