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?