### subgraph

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

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?

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.

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.

An undirected graph homomorphism h: H -> G is said to be a monomorphism when h on vertices is an injective function. As a graph homomorphism h of course maps edges to edges but there is no requirement that an edge h(v0)-h(v1) is reflected in H. The case of directed graphs is similar.

There is a discrepancy between Math and CS terms here. From math you get two terms: subgraph isomorphism: Let H = (VH, EH) and G = (V, E) be graphs. f : VH → V is a subgraph isomorphism if (u, v) ∈ EH, then (f(u), f(v)) ∈ E. H is isomorphic to a subgraph of G induced subgraph isomorphism: Let H = (VH, EH) and G = (V, E) be graphs. f : VH → V is an induced subgraph isomorphism if (u, v) ∈ EH, then (f(u), f(v)) ∈ E. And if (u, v) is not and element of EH, then (f(u), f(v)) is not an element of E. H is isomorphim to an induced subgraph of G Definitions from http://theory.stanford.edu/~virgi/cs267/lecture1.pdf. They are equivalent to what I found in "A First Course in Graph Theory." Note that both of these are injective homomorphisms between graphs aka a graph monomorphism. Moving to CS and specifically the subgraph isomorphism problem. To the best of my understanding a subgraph isomorphism algorithm determines if a function exists that satisfies (2) from above. Graph monomorphism matches (1). The CS definitions are from the VF2 algorithm, I do not know how widespread that usage is. While searching for the correct algorithm for a project it seems like there is still some ambiguity and not all projects use the same definitions. Take this answer with a grain of salt http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.101.5342&rep=rep1&type=pdf lists monomorphism as seperate from graph-subgraph isomorphism in the introduction, but in section 2 defines graph-subgraph isomorphism as conceptually identical to (1) which indicates I am missing something.

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