We investigate a combinatorial reconfiguration problem on oriented graphs, where a reconfiguration step (edge-flip) is the inversion of the orientation of a single edge. A recently published conjecture that is relevant to the correctness of a Markov Chain Monte Carlo sampler for directed flag complexes states that any simple oriented graph admits a flip sequence that monotonically decreases the number of directed 3-cycles to zero, and is known to be true for complete oriented graphs. We show that, in general, this conjecture does not hold. As main tool for disproving the conjecture, we introduce the concept of FBD-graphs (fully blocked digraphs). An FBD-graph is a directed graph that does not contain any directed 1-, 2-, or 3-cycles, and for which any edge-flip creates a directed 3-cycle. We prove that the non-existence of FBD-graphs is a necessary condition for the conjecture to hold and succeed in constructing FBD-graphs. On the other hand, we show that complete graphs, as well as graphs in which every edge is incident to at most two triangles, cannot be fully blocked. In addition to being relevant for determining in which cases the above mentioned sampling process is correct, the concept of FBD-graphs might also be useful for other problems and yields interesting questions for further study.

翻译：我们研究有向图上的组合重配置问题，其中重配置步骤（边翻转）指反转单条边的方向。一个最近发表的、与有向旗复形马尔可夫链蒙特卡洛采样器正确性相关的猜想声称：任何简单有向图都存在一个翻转序列，能单调地将有向3-环的数量减少至零，且该猜想已知在完全有向图上成立。我们证明该猜想在一般情况下不成立。作为证伪猜想的主要工具，我们引入了FBD图（全阻塞有向图）的概念。FBD图是一种不包含任何有向1-环、2-环或3-环的有向图，且其任意边翻转都会产生一个有向3-环。我们证明了FBD图的不存在性是猜想成立的必要条件，并成功构造了FBD图。另一方面，我们证明完全图以及每条边最多关联两个三角形的图不可能被完全阻塞。FBD图的概念除了有助于确定上述采样过程在何种情况下正确外，也可能对其他问题具有应用价值，并为进一步研究提供了有趣的问题。