Fix typo in Strongly Connected Components article (#159)

joaquingx · tcNickolas · commit a918550b3e94 · 2017-06-06T12:43:51.000-07:00
diff --git a/src/graph/strongly-connected-components.md b/src/graph/strongly-connected-components.md
@@ -27,7 +27,7 @@ First, let's make notations: let's define exit time $tout[C]$ from the strongly
 **Theorem**. Let $C$ and $C'$ are two different strongly connected components and there is an edge $(C, C')$ in a condensation graph between these two vertices. Then $tout[C] > tout[C']$.
 
 There are two main different cases at the proof depending on which component will be visited by depth first search first, i.e. depending on difference between $tin[C]$ and $tin[C']$:
-** The component $C$ was reached first. It means that depth first search comes at some vertex $v$ of component $C$ at some moment, but all other vertices of components $C$ and $C'$ were not visited yet. By condition there is an edge $C, C'$ in a condensation graph, so not only the entire component $C$ is reachable from $v$ but the whole component $C'$ is reachable as well. It means that depth first search that is running from vertex $v$ will visit all vertices of components $C$ and $C'$, so they will be descendants for $v$ in a depth first search tree, i.e. for each vertex $u \in C \cup C', u \ne v$ we have that $tout[v] > tout[u]$, as we claimed.
+** The component $C$ was reached first. It means that depth first search comes at some vertex $v$ of component $C$ at some moment, but all other vertices of components $C$ and $C'$ were not visited yet. By condition there is an edge $(C, C')$ in a condensation graph, so not only the entire component $C$ is reachable from $v$ but the whole component $C'$ is reachable as well. It means that depth first search that is running from vertex $v$ will visit all vertices of components $C$ and $C'$, so they will be descendants for $v$ in a depth first search tree, i.e. for each vertex $u \in C \cup C', u \ne v$ we have that $tout[v] > tout[u]$, as we claimed.
 ** Assume that component $C'$ was visited first. Similarly, depth first search comes at some vertex $v$ of component $C'$ at some moment, but all other vertices of components $C$ and $C'$ were not visited yet. But by condition there is an edge $(C, C')$ in the condensation graph, so, because of acyclic propery of condensation graph, there is no back path from $C'$ to $C$, i.e. depth first search from vertex $v$ will not reach vertices of $C$. It means that vertices of $C$ will be visited by depth first search later, so $tout[C] > tout[C']$. This completes the proof.
 
 Proved theorem is **the base of algorithm** for finding strongly connected components. It follows that any edge $(C, C')$ in condensation graph comes from a component with a larger value of $tout$ to component with a smaller value.
