# homeomorphic graphs

Are these graphs homeomorphic?

a) and the Peterson graph?

b) I think both are not homeomorphic.Is it correct?Is there a way I can show that they are not homemorphic
Also two graphs being isomorphic doesn't imply they are homeomorphic right?

• Do you mean isomorphic? Oct 10, 2014 at 15:24
• Homeomorphic is a topological notion, and isomorphic is an algebraic notion, I was wondering which you meant. Oct 10, 2014 at 15:27
• Homeomorphism is a concept in graph theory as well. Oct 10, 2014 at 15:31
• Some authors use "graph homeomorphism" to mean the same thing as "graph isomorphism". However the Wikipedia article Graph homeomorphism defines it to be a more general relation, namely graphs $G$ and $G'$ are homeomorphic iff subdivisions of $G$ and $G'$ may be found which are isomorphic. [A subdivision of a (simple undirected) graph inserts new nodes on edges.] Oct 10, 2014 at 16:05

If by graph homeomorphisms we mean the isomorphisms of graph subdivisions (isomorphism after introducing new nodes that subdivide one or more edges), then a necessary (but not always sufficient) criterion asks if the reduced degree sequences of the two graphs (meaning that degree $2$ entries are deleted from the degree sequences) are the same.
• I think a better way to express graph homeomorphism of $G$ and $G'$ is to say they may be made isomorphic by a series of graph subdivisions (rather than saying each may be obtained from the same graph by the process of subdividing edges). Oct 10, 2014 at 21:38