On independent circuits of a digraph

Abstract

If every three circuits of a digraph have a common vertex, then all the circuits have one.Paths and circuits are assumed to be elementary, directed, and of nonzero length. A set of paths is openly disjoint if every pair of paths in the set is vertex disjoint with the possible exception that the starting vertex of one could be the same as the final vertex of the other, if neither path is a circuit. A vertex u whose removal makes a digraph G acyclic is a break vertex of G. A carrier [l, p. 30; 7, p. 171 is a vertex with one in-arc and one out-arc. A digraph is minimally connected if it is a strong digraph and the removal of any arc destroys the strongly connected property. A digraph is arbitrarily traceable from u if each chain obtained by starting from u, and choosing at each vertex u an out-arc of u which has not been previously traversed, is closed and complete, A digraph having a closed complete chain is eulerian.Harary [S] gave a characterization of eulerian digraphs with a break vertex: An eulerian digraph G has a break vertex u if and only if G is arbitrarily traceable from u. Here a characterization of digraphs having a break vertex is given.The following two lemmas are minor variants of well-known results.Lemma 1 [l, p. 311. Every minimally connected digraph has a carrier.

On structuring flowcharts (Preliminary Version)

“…If every 3 circuits of G have a common vertex, then a single vertex is on every circuit of G [9]. Any such vertex blanks every circuit in G.…”

An Eulerian exposition

“…In general, analogous results hold for multidigraphs, with two interesting exceptions. First, Kosaraju [32] showed that in a strong multidigraph D, the property that some vertex lies on every cycle of D is equivalent to the property that every three cycles of D have some common vertex. This, together with a result of Harary, provides us with two characterizations of multidigraphs which are randomly Eulerian from some vertex.…”