So far, I have found out that chordal graphs have linear number of maximal cliques with respect to the number of vertices. In general case, it is exponential.
I am trying to determine whether the number of maximal cliques in a $(2C_4, C_5,P_5)$-free graph is linear or polynomial with respect to the number of vertices.
In a $(2C_4, C_5,P_5)$-free graph, the largest induced cycle is of length 4, and no two 4-cycles are edge-disjoint.
Is there a paper that mentions such result?
(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
