Maximal strongly connected cliques in directed graphs: Algorithms and bounds

Finding communities in the form of cohesive subgraphs is a fundamental problem in network analysis. In domains that model networks as undirected graphs, communities are generally associated with dense subgraphs, and many community models have been proposed. Maximal cliques are arguably the most widely studied among such models, with early works dating back to the ’60s, and a continuous stream of research up to the present. In domains that model networks as directed graphs, several approaches for community detection have been proposed, but there seems to be no clear model of cohesive subgraph, i.e., of what a community should look like. We extend the fundamental model of clique to directed graphs, adding the natural constraint of strong connectivity within the clique. We consider in this paper the problem of listing all maximal strongly connected cliques in a directed graph. We first investigate the combinatorial properties of strongly connected cliques and use them to prove that every n-vertex directed graph has at most 3n∕3 maximal strongly connected cliques. We then exploit these properties to produce the first algorithms with polynomial delay for enumerating maximal strongly connected cliques: a first algorithm with polynomial delay and exponential space usage, and a second one, based on reverse-search, with higher (still polynomial) delay but which uses linear space.1