The worst-case time complexity for generating all maximal cliques and computational experiments

Abstract

We present a depth-first search algorithm for generating all maximal cliques of an undirected graph, in which pruning methods are employed as in the Bron–Kerbosch algorithm. All the maximal cliques generated are output in a tree-like form. Subsequently, we prove that its worst-case time complexity is $\mathrm{O}(3^{n/3})$ for an n-vertex graph. This is optimal as a function of n, since there exist up to $3^{n/3}$ maximal cliques in an n-vertex graph. The algorithm is also demonstrated to run very fast in practice by computational experiments.