Isomorphic, Homeomorphic, and Subgraphs in Graph Theory

Explore isomorphic, homeomorphic, and subgraph relationships in graph theory. This guide defines these concepts, explains their differences, and provides illustrative examples to clarify these important graph relationships.

Isomorphic, Homeomorphic, and Subgraphs

Isomorphic Graphs

Two graphs, G and G*, are isomorphic if they have the same structure. This means there's a one-to-one correspondence (a way to match up the vertices of G with the vertices of G*) such that if two vertices are connected in G, the corresponding vertices are connected in G*, and vice versa. Essentially, isomorphic graphs are just different drawings of the same underlying graph.

Homeomorphic Graphs

Two graphs, G and G*, are homeomorphic if they can both be obtained from the same graph (or from isomorphic graphs) by adding vertices of degree 2 along the edges. Homeomorphic graphs have the same structure ignoring the vertices of degree 2.

(Illustrative figures showing two non-isomorphic but homeomorphic graphs and the common base graph would be included here.)

Subgraphs

A subgraph G' of a graph G is a smaller graph contained within G. This means that all the vertices and edges of G' are also in G, and the connections between vertices in G' are the same as in G.

Example: Subgraphs

(An illustrative graph G would be shown here, followed by several smaller graphs that are subgraphs of G. A note that a single vertex is also considered a subgraph would be included.)

Spanning Subgraphs

A spanning subgraph G' of a graph G is a subgraph that includes all the vertices of G. It might not include all the edges of G, but it must contain all the vertices.

Example: Spanning Subgraph

(An illustrative graph G would be shown here, followed by a spanning subgraph of G. The emphasis should be that the subgraph has all vertices of G but possibly fewer edges.)

Conclusion

Isomorphism, homeomorphism, and subgraph relationships are fundamental concepts in graph theory, providing ways to compare and analyze the structure of different graphs.