Dec 9, 2024 · Such an f is called an isomorphism of the graphs G and G′. While the definition seems to consider the set of vertices instead of labels, I’m confused because many textbook and internet examples express graph isomorphism as a map from labels to labels just like the below picture from Wikipedia.

Feb 13, 2018 · Two connected 2-regular graphs with countable infinite many vertices are always isomorphic. This graph is called double-ray. There is a model of random graphs on a countable infinite set of vertices such that every such graph is isomorphic to any other. This graph is called the Rado graph.

Oct 24, 2017 · A more general approach to graph isomorphism is to look for graph invariants: properties of one graph that may or may not be true for another. (The degree sequence of a graph is one graph invariant, but there are many others.) This is usually a quick way to prove that two graphs are not isomorphic, but will not tell us much if they are. For ...

Jun 22, 2015 · In addition to the above-mentioned definition of a graph invariant, I would like to point out that, according to some authors, a graph invariant may be something different from a number; for example, according to Gross and Yellen (Graph Theory and its Applications, Second Edition, CRC, Boca Raton, 2006, page 89), "a graph invariant is a property of graphs that is …

Jun 1, 2020 · I have one question about the algorithm on testing the Trivalent Graph isomorphism in polynomial time. The paper "Isomorphism of graphs of bounded valence can be tested in polynomial time" by Luks was complicated to me. As I understood, given 2 Trivalent Graphs, we can test whether they are isomorphism in polynomial time.

BTW, your 2. is an excerpt from Wikipedia entry Graph Automorphism, if you'd have bothered to read the next sentence you would see: "...That is, it is a graph isomorphism from G to itself." $\endgroup$

Isomorphism is the stringier constraint, since it requires a bijection between the vertices and edges from one graph to another such that vertex adjancencies are preserved. Equality, on the other hand, only requires that two graphs have the same number of vertices and edges.

This will give you an isomorphism. How did I get here: Well an isomorphism is a relation that preserves vertex adjacency in two graphs. Examining the definition properly you will understand that two graphs are isomorphic implies vertices in both graphs are adjacent to each other in the same pattern. So the geometric picture of a graph is useless.

Mar 10, 2019 · Any one-to-one correspondence between the vertices of one graph and the vertices of another graph is a candidate for an isomorphism --- a successful candidate if the edges then also match up. Travis's answer has given you an appropriate correspondence between the pentagon and the $5$ -pointed star.

In other words, proofs of NP-completeness seem to require a certain amount of redundancy in the target problem, a redundancy that GRAPH ISOMORPHISM lacks. Unfortunately, this lack of redundancy does not seem to be much of a help in designing a polynomial time algorithm for GRAPH ISOMORPHISM either, so perhaps it belongs to NPI.