# How to denote a graph class which allows only $k$ instances of a certain induced subgraph?

Suppose that a graph class $$\mathcal{C}$$ is defined as follows:

A graph $$G$$ belongs to $$\mathcal{C}$$ if, and only if $$G$$ is chordal, but has at most $$k$$ $$5$$-cycles.

I am aware that the definition of $$\mathcal{C}$$ is contradictory. However, I am looking for a notation like

$$G \in C_4\text{-free} + k*(C_5)$$

Is there such notation in the literature?

• Not a proper answer, but maybe a start: collections of graphs from classes and sets, so set notation is entirely applicable, for example: graphclasses.org/classes.cgi – Luke Mathieson Nov 28 '18 at 23:33
• @LukeMathieson That is true, but the set notation has no limitations on the number of instances. – padawan Nov 29 '18 at 0:46