e_maxx_link: strong_connected_components

Let $G=(V,E)$ be a directed graph with vertices $V$ and edges $E$. We denote with $n=|V|$ the number of vertices and with $m=|E|$ the number of edges in $G$.

<img src="strongly-connected-components-tikzpicture/graph.svg" alt="drawing" style="width:700px;"/>

We define the **condensation graph** $G^{\text{SCC}}=(V^{\text{SCC}}, E^{\text{SCC}})$ as follows:

- the vertices of $G^{\text{SCC}}$ are the strongly connected components of $G$; i.e., $V^{\text{SCC}} = \text{SCC}(G)$, and

- for all vertices $C_i,C_j$ of the condensation graph, there is an edge from $C_i$ to $C_j$ if and only if $C_i \neq C_j$ and there exist $a\in C_i$ and $b\in C_j$ such that there is an edge from $a$ to $b$ in $G$.

<img src="strongly-connected-components-tikzpicture/cond_graph.svg" alt="drawing" style="width:700px;"/>