Let $G$ be a finite directed graph with vertex set $V$ and an edge set $E$. In this paper https://arxiv.org/pdf/1209.2578.pdf, another graph is constructed from $G$ called $R$-graph by the following way:
- On page 2, line-5 of the paper whose link is given above, Authors choose a partition $\{E^{-}, E^+\}$ of $E$.
- In page 2, line-13, for any $q, r \in V$, Authors choose a relation $\mathcal R(q, r) \subset E^{-}(q, r) \times E^+(q, r)$
I have doubts about this partition and relation. Whether this partition $\{E^{-}, E^+\}$, and relation $\mathcal R(q, r)$ are fixed or for any partition $\{E^{-}, E^+\}$ and relation $\mathcal R(q, r)$, we can construct $R$-graph?
Thanks in advance for your kind help.
Krieger is simply defining his notion of $\mathcal{R}$-graph. It is an object $\mathcal{G}_{\mathcal{R}}(\mathfrak{P},\mathcal{E}^-,\mathcal{E}^+)$, where $\mathfrak{P}$ is the vertex set of a finite directed graph with edge set $\mathcal{E}$, $\{\mathcal{E}^-,\mathcal{E}^+\}$ is a partition of $\mathcal{E}$, the graph satisfies certain conditions, and $\mathcal{R}$ is a relation on the edges that also satisfies certain conditions.
Specifically, for $\mathfrak{q},\mathfrak{r}\in\mathfrak{P}$ we define $\mathcal{E}^-(\mathfrak{q},\mathfrak{r})$ to be the set of edges in $\mathcal{E}^-$ from $\mathfrak{q}$ to $\mathfrak{r}$ and $\mathcal{E}^+(\mathfrak{q},\mathfrak{r})$ to be the set of edges in $\mathcal{E}^+$ from $\mathfrak{r}$ to $\mathfrak{q}$. For each $\mathfrak{q},\mathfrak{r}\in\mathfrak{P}$ we require that either both of these sets be empty or both be non-empty, and we further require that the directed graph $\langle\mathfrak{P},\mathcal{E}^-\rangle$ (and hence also ($\langle\mathfrak{P},\mathcal{E}^+\rangle$) be strongly connected.
The relation $\mathcal{R}$ is the union of relations $\mathcal{R}(\mathfrak{q},\mathfrak{r})\subseteq\mathcal{E}^-(\mathfrak{q},\mathfrak{r})\times\mathcal{E}^+(\mathfrak{q},\mathfrak{r})$ for $\mathfrak{q},\mathfrak{r}\in\mathfrak{P}$.
And that’s it: any object satisfying those conditions is an $\mathcal{R}$-graph in the sense of this definition. If you change any of the parameters — the vertex set $\mathfrak{P}$, the edge set $\mathcal{E}$, the partition of the edge set, or the relation $\mathcal{R}$ — you have either a different $\mathcal{R}$-graph, if the various structural conditions are still met, or something that is not an $\mathcal{R}$-graph.