On support sets of acyclic and transitive digraphs

In previous works of the authors, the concept of a binary reflexive adjacency relation was introduced on the set of all binary relations of the set $X$, and an algebraic system consisting of all binary relations of the set $X$ and of all unordered pairs of adjacent binary relations was defined. If $X$ is a finite set, then this algebraic system is a graph (graph of binary relations $G$). The current paper introduces the notion of a support set for acyclic and transitive digraphs. This is the collections $S(\sigma)$ and $S'(\sigma)$ consisting of the vertices of the digraph $\sigma\in G$ that have zero indegree and zero outdegree, respectively. It is proved that if $G_\sigma $ is a connected component of the graph $G$ containing the acyclic or transitive digraph $\sigma\in G$, then $\{S(\tau): \tau\in G_\sigma\}=\{S'(\tau): \tau\in G_\sigma\}$. A formula for the number of transitive digraphs having a fixed support set is obtained. An analogous formula for the number of acyclic digraphs having a fixed support set was obtained by the authors earlier.