What's the Difference Between Subgraph Isomorphism and Subgraph Monomorphism?

Question

In one of the projects I've worked on, the subject of isomorphism versus monomorphism came up.

A little background: I'm no expert on graph theory and have no formal training in it. But this topic is very important in chemistry, where chemists expect a particular kind of subgraph matching to take place in the structure search systems they use.

If a target graph A has n nodes and m edges, then a chemist would accept subgraph matches in which a query graph B had n nodes and m-1 edges. The only requirement would be that every edge in B should be present in A. For example, a linear chain of 6 nodes should match a cycle of 6 nodes.

Is this kind of matching isomorphism or monomorphism? Maybe something else altogether?

Answer 1

Let G1, G2 be graphs composed of sets of vertices and edges V1, V2 and E1, E2 respectively.

G2 is isomorphic to a subgraph of G1 iff there exists a one-one mapping between each vertex of V2 and a vertex in V1, and between each edge in E2 and some edge in E1. So to be isomorphic, you need to have an exact match, including if the graph includes more than one edge between nodes.

G2 is monomorphic iff there's such a mapping between vertices, and there exists an edge between all vertices in V2 for which there is a corresponding edge in V1. But so long as at least one edge exists, that's sufficient.

Here's a nice package description from comp.lang.python.

Answer 2

Monomorphism.

A monomorphism from one graph ("B") to another graph ("A") is equivalent to an isomorphism from B to a subgraph of A.

The example is saying that any n vertex path ("chain") is monomorphic to an n vertex cycle. It would also be monomorphic to a n+1 vertex cycle, or n+k for any k.