Clique Enumeration Algorithm I am reading an old paper of M.C. Golumbic about EPT (edge intersection of paths in a tree) graphs. In the paper it is shown that the number of maximal cliques of an EPT graph instance is polynomial. It concludes that if an oracle reports that a graph $G$ is an EPT graph, then  it is possible to find the maximum clique with a standard clique enumeration algorithm.

First of all, what are these standard clique enumeration algorithms? If there are more than one, can we say that if the number of maximal cliques of a graph is polynomial then can we use any of these enumeration algorithms? Or should we derive a special algorithm from a generic algorithm which uses some special structures of the graph class?

Thanks in advance.
There are several output-sensitive algorithms to enumerate all maximal cliques in polynomial time per output.  One of the earliest algorithms was developed by Tsukiyama, Ide, Ariyoshi, and Shirakawa (1977).


Shuji Tsukiyama, Mikio Ide, Hiromu Ariyoshi, Isao Shirakawa: A New Algorithm for Generating All the Maximal Independent Sets. SIAM J. Comput. 6(3): 505-517 (1977)


This means that if you know your graph has at most polynomially many maximal cliques, then the total running time of their algorithm will be polynomial in the input size.