Cycles in directed graphs

Question

Let G be a finite directed graph (allowing multiple edges). We define a cycle (as usual) to be a sequence of edges $e_0, e_1, \dots, e_{n-1}$ (up to cyclic permutation) such that the terminal vertex of $e_i$ is the initial vertex of $e_{i+1}$ (indexing modulo $n$). A cycle is minimal if no vertex appears more than once among the initial vertices of the sequence.

We can assume that $G$ is strongly connected, that is, there is a directed path between any two vertices. For my purposes, we can also assume that every vertex has at least two out-going edges and at least two incoming edges (in fact, we can reduce to this).

Finally, here is the question. Does there exist such a directed graph such that for every edge $e$, whenever $D$ and $E$ are distinct minimal cycles containing $e$, there exists a third (distinct) minimal cycle $C$ with $C \subset D \cup E$ and $e$ not in $C$. [That $D \cup E$ even contains a third minimal cycle is fairly strong on its own.]

The question arises in the classification of dynamical systems on zero-dimensional spaces, and also in the (ordered) cohomology of graphs. (This requires a chain of explanations.)

Answer 1

The proposition is true vacuously for a directed cycle, or a directed graph in which every biconnected component is a cycle. In order to get such a graph and also meet your assumptions on the minimum indegree and out-degree, one would have to allow self-loops.

For any other graph, it's false. For, let $G$ be biconnected (otherwise split into components and treat each one separately) and not itself a single cycle. Then $G$ has a directed open ear decomposition. Let $E$ be the first ear (a simple directed cycle) and $F$ be the second ear (a simple path starting and ending at different vertices of $E$, and otherwise disjoint from $E$). Let $D$ be the cycle consisting of $F$ together with the path in $E$ connecting its endpoints. Then $D$ and $E$ share at least one edge $e$, but there is no third minimal cycle in $D\cup E$.