Today I start a small series of posts that touch on the different approaches "available in the market" to graph dynamics. More material can be found in chapter 3 of the MGG book, and the reader is referred there for the missing details. Let's start saying that there is a very active community made up mainly of computer scientists dedicated to this topic. The ideas behind them are deep and the results derived after over almost 40 years of research very interesting.

Nowadays, no doubt, the most important approach to graph transformation is the one known as

*algebraic approach*. However, I will refer to it as

*categorical approach*because it uses category theory. There is no algebra in it. In fact, it is not a single approach: At least there exist the single pushout (SPO) and the double pushout (DPO) approaches, and a generalization known as

*adhesive HLR categories*(AHLRC). A good book on the topic is this one.

The main advantage of using category theory, in principle, is that once a result is established, it applies to any mathematical structure that belongs to that category. However, DPO and SPO should not be thought of as theories that study a concrete category, but rather as using a categorical construction to define the application of a production to a concrete host graph (this is known as a

*derivation*). In particular, SPO uses a pushout construction and DPO uses two pushout constructions. (This is explained in detail in pp. 22 and 23 and 41-49 in the MGG book.) The one that does study categories to which results can be extended is AHLRC. As far as I know, AHLRC only deals with DPO and in general does not apply to SPO.

The next post is about what I see as advantages and "problems" of SPO and DPO. It is just my personal view

I quote Alfred North Whitehead to finish today's post:

**Everything of importance has been said before by somebody who did not discover it**.

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