Short cycle connectivity

Short cycle connectivity is a generalization of ordinary connectivity. Instead of by a path (sequence of edges), two vertices have to be connected by a sequence of short cycles, in which two consecutive cycles have at least one common vertex. If all consecutive cycles in the sequence share at least one edge, we talk about edge short cycle connectivity. Short cycle connectivity can be extended to directed graphs (cyclic and transitive connectivity). It is shown that the short cycle connectivity is an equivalence relation on the set of vertices, while the edge/arc short cycle connectivity components determine an equivalence relation on the set of edges/arcs. Some additional properties of these relations are also presented. The related notion of short cycle networks provides us with a tool for identification of dense parts of graphs with applications in the design of algorithms and social network analysis (hierarchies, Granovetter’s strong and weak ties). For further generalization we can also consider connectivity by small cliques or other families of graphs.