Number of maximal cliques in a ($2C_4$, $C_5$, $P_5$)-free graph
(Answered also on https://cstheory.stackexchange.com/questions/47691/)
A ($2C_4$, $C_5$, $P_5$)-free graph may have exponentially many maximal cliques.
For example, the complement of the disjoint union of $n/3$ triangles with $3^{n/3}$ maximal cliques is $K_1 \cup K_2$-free, and thus has none of $2C_4$, $C_5$, $P_5$ appear as an induced subgraph.
https://doi.org/10.1007/BF02760024
Note: If the complement of a graph has $k$ pairwise independent edges, then they give you $2^k$ maximal cliques. Conversely, it is known that the number of maximal cliques is upper-bounded by a function of the maximum number of independent edges in the complement .
https://onlinelibrary.wiley.com/doi/abs/10.1002/net.3230230308