A Note on an Old-Fashioned Algebra for (Disconnected) Graphs

Graphs with interfaces are a simple and intuitive tool for allowing a graph G to interact with the environment, by equipping it with two morphisms J → G, I → G. These “handles” were used to define graphical operators, and to provide an inductive presentation of graph rewriting. A main feature of graphs with interfaces is their characterization as terms of a free algebra. So far, this was possible only with discrete interfaces, i.e., containing no edge. This note shows that a similar free construction can be performed also with disconnected interfaces, i.e., containing only nodes connected to at most one edge.