I'm working in the category of simple graphs, where objects are undirected graphs with loops at every vertex and multiple edges are forbidden, and morphisms are graph morphisms, that are edge-preserving functions of vertices.
It is well known that decomposition into connected components, e.g. decomposition of $G$ into maximum number of graphs $\{G_i\}$ such that $G = G_1 \sqcup \dotsb \sqcup G_n$, is highly useful, because we can treat $G$ as collection of graphs, so for example we can use handshake lemma not on $G$ but on each $G_i$ and so on.
And while categorical definition of coproduct uses generic morphisms, we only need to know that there are paths $P_m \hookrightarrow G_i$ with some properties.
So I was wondering:
- Is there any useful properties of categorical product, known as strong product (if I'm right)?
- How to simplify it's identification, e.g. what do I need to know about graph $G$ so I can decompose it to form $G \cong G_1 \times G_2$?
- Is there any other categorical constructions that are useful in terms of graph theory?