Listing all maximal cliques in sparse graphs in near-optimal time.

The degeneracy of an n-vertex graph G is the smallest number d such that every subgraph of G contains a vertex of degree at most d. We show that there exists a nearly-optimal fixed-parameter tractable algorithm for enumerating all maximal cliques, parametrized by degeneracy. To achieve this result, we modify the classic Bron-Kerbosch algorithm and show that it runs in time O(dn3(d/3)). We also provide matching upper and lower bounds showing that the largest possible number of maximal cliques in an n-vertex graph with degeneracy d (when d is a multiple of 3 and n >= d + 3) is (n - d)3(d/3). Therefore, our algorithm matches the Theta(d(n - d)3(d/3)) worst-case output size of the problem whenever n - d = Omega(n).