On Some Cyclic Connectivity Properties of Directed Graphs

  (Examples and Problems) 1

A. Ádám 3

To Professor Ferenc Gécseg on his sixtieth birthday

Introduction

 The essence of the paper consists in ten properties (each defining a class of finite directed graphs) listed in § 2 and in open questions (relating to dependencies among the properties) raised in §§ 8-10.

 A number of dependence and independence assertions can be deduced easily or follow trivially from the ten properties. The originality of the statements in §§ 3, 8, 9 and of the examples in §§ 6, 7 does not exceed the level of routine consequences of the definitions.

 Since the number of properties is ten, one can think ``a priori" that the class of graphs which possess at least one property is partitioned into 1023 ( = 210 -1) subclasses (called types). In fact, the dependency statements imply that there are not more than twenty-one types; on the other hand, examples are got for ten types. The 21 imaginable types correspond in a natural manner to the 21 independent vertex sets of the hierarchy diagram shown in Figure 1. One of the types consists of some connected graphs which are not strongly connected, the remaining Ł 20 types constitute a partition of the class of the strongly connected graphs.

 A part of the open problems concerns to the existence of the eleven types whose non-emptiness is not decided in the article. In the last section an exciting topics is affected: the (fond?) hope for elaborating a structure theory of the strongly connected (directed) graphs.

Footnotes:

  1 Research partially supported by the Hungarian National Foundation for Scientific Research (OTKA) grant, no. T 16389.

  2 Research partially supported by the Hungarian National Foundation for Scientific Research (OTKA) grant, no. T 16389.

  3 MTA Matematikai Kutatóintézet, H-1364 Budapest, P.O.Box 127., Hungary.