Untangling Multigraphs from Pseudographs: A Friendly Guide
A Gentle Note on Terminology
While multigraphs are quite open to the idea of loops, it's worth briefly touching upon a related term you might come across: the pseudograph. The distinction, while perhaps appearing subtle, is quite important in the formal language of graph theory. A pseudograph is essentially a multigraph that is explicitly given permission to have loops. Now, you might be thinking, "Wait, didn't we just establish that multigraphs can have loops?" You are absolutely right, and here's where the delightful nuance comes in.
In some definitions, especially those that are older or more rigorously traditional, a "multigraph" might have been understood to refer only to graphs with multiple edges but strictly *without* loops. In these specific contexts, the term "pseudograph" was introduced specifically to denote a graph that permits both multiple edges and loops. However, in the contemporary landscape of graph theory, the more widely accepted and commonly used definition of a "multigraph" *does* generously include the possibility of loops. This means that a pseudograph, in most modern interpretations, is simply a specific, perhaps slightly more emphasized, type of multigraph.
It's a bit like how a square is a special kind of rectangle. All squares are rectangles, but not every rectangle has the precise characteristics to be a square. Similarly, in many current definitions, all pseudographs are indeed multigraphs, but not all multigraphs are necessarily called pseudographs if one were to use a definition where "multigraph" implicitly meant *no* loops (though, I must reiterate, this interpretation is becoming increasingly uncommon).
The most important takeaway is this: if you encounter the term "pseudograph," understand that it unequivocally signifies a graph that warmly embraces both multiple edges between points and the charming presence of loops. When we discuss "multigraphs" today, the prevailing understanding is that loops are simply part of the package, a welcome addition. It's always a good idea to gently check the specific definitions being used in any given textbook or academic work to avoid any tiny misunderstandings—graph theory, bless its heart, can sometimes be quite particular about its precise language!
