Cyclic Connectivity Classes of Directed Graphs

We show that most connectivity types of simple directed graphs defined by A. Ádám in [1] are nonempty. The nonexistence of the three types that remain missing is the consequence of a fairly plausible conjecture, stated at the end of this paper. Introduction This paper gives an almost complete answer to a question raised by A. Ádám in [1] concerning the connectivity types of simple directed graphs. In his paper A. Ádám defined ten properties pertinent to the cyclic structure of a simple directed graph. He also explored most (possibly all) implications between these properties, resulting in a hierarchy of cyclic connectivity, represented on Figure 1. By studying the acyclic directed graph of logical dependencies he concluded that his proposed classification may yield at most twenty one disjoint types of cyclic connectivity for directed graphs. He also constructed examples for ten types and raised the question whether all other eleven types are nonempty. The main result of this paper is that at least eight of these missing types is not missing any more. Furthermore, the techniques used to construct our examples reveal interesting connections between Ádám’s cyclic connectivity classes, some of which are either invariant or go consistently into the same other class under the effect of such simple operations as contracting edges or expanding vertices. These operations may, by the way, yield a graph that is not simple any more but we indicate at least one standard method –the use of “compasses”– to get around this issue. This technique hints how to extend Ádám’s theory to non-simple directed graphs as well. We conclude our paper with an important conjecture which, if true, would explain why the remaining three missing types are still missing: our conjecture implies that no graph of these types would exist. Our conjecture, if true may reveal a deep interrelation between the cyclic connectivity properties of directed graphs. 1. Basic concepts and facts We define a directed graph G as a pair (V,E) of a finite nonempty vertex set V and of a subset E of V × V . We consider only graphs which are simple in the following sense: they contain no loops (edges of the form (v, v)) and there is at most one edge between any fixed pair of vertices. Furthermore, we denote the indegree resp. outdegree (the number of incoming resp. outgoing edges) of a vertex a by 2000 Mathematics Subject Classification. Primary 05C20; Secondary 05C38, 05C40.