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 $O(dn{3}^{d\mathrm{/}3})$. 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 $n\ge d+3$) is $(n-d)3^{d\mathrm{/}3}$. Therefore, our algorithm matches the $\mathrm{\Theta }(d(n-d)3^{d\mathrm{/}3})$ worst-case output size of the problem whenever $n-d=\mathrm{\Omega }(n)$.