# reference query – How to designate a class of graphs allowing only \$ k \$ instances of a certain induced subgraph?

Suppose that a class of graphs $$mathcal {C}$$ is defined as follows:

A graph $$G$$ belongs to $$mathcal {C}$$ if and only if $$G$$ is in agreement,
but 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 a notation in the literature?