Update Silver_Graphs.md

content/2_Silver/Silver_Graphs.md

@@ -71,6 +71,8 @@ The most common example of a two-colored graph is a *bipartite graph*, in which
 The idea is that we can arbitrarily label a node and then run DFS. Every time we visit a new (unvisited) node, we set its color based on the edge rule. When we visit a previously visited node, check to see whether its color matches the edge rule. For example, an implementation of coloring a bipartite graph is shown below.
 
 ```cpp
+//UNTESTED
+
 bool is_bipartite = true;
 void dfs(int node)
 {
@@ -101,6 +103,8 @@ void dfs(int node)
 
 A *cycle* is a non-empty path of distinct edges that start and end at the same node. *Cycle detection* determines properties of cycles in a directed or undirected graph, such as whether each node of the graph is part of a cycle. 
 
+A related topic is **strongly connected components**, a platinum level concept.
+
 ### Functional Graphs
 
 Links:
@@ -112,13 +116,13 @@ In silver-level directed cycle problems, it is generally the case that each node
 
 The following sample code counts the number of cycles in such a graph. The "stack" contains nodes that can reach the current node. If the current node points to a node `v` on the stack (`on_stack[v]` is true), then we know that a cycle has been created. However, if the current node points to a node `v` that has been previously visited but is not on the stack, then we know that the current chain of nodes points into a cycle that has already been considered.
 
-(test code?)
-
 ```cpp
+//UNTESTED
+
 //Each node points to next_node[node]
 
 bool visited[MAXN], on_stack[MAXN];
-int number_of_cycles = 0;
+int number_of_cycles = 0, next_node[MAXN];
 void dfs(int n)
 {
 	visited[n] = on_stack[n] = true;
@@ -138,7 +142,51 @@ int main()
 }
 ```
 
-For non-functional directed graphs, this code will still detect a cycle if it exists (though `number_of_cycles` will be meaningless).
+The same general idea is implemented below to find any cycle in a directed graph.
+
+```cpp
+//UNTESTED
+
+bool visited[MAXN], on_stack[MAXN];
+vector<int> adj[MAXN];
+vector<int> cycle;
+bool dfs(int n)
+{
+	visited[n] = on_stack[n] = true;
+	for(int u:adj[n])
+	{
+		if(on_stack[u])
+			return cycle.push_back(v), cycle.push_back(u), on_stack[n] = on_stack[u] = false, true;
+		else if(!visited[u])
+		{
+			if(dfs(u))
+				if(on_stack[n])
+					return cycle.push_back(n), on_stack[n] = false, true;
+				else
+					return false;
+			if(!cycle.empty())
+				return false;
+		}
+	}
+	on_stack[n] = false;
+	return false;
+}
+int main()
+{
+	//take input, etc
+	for(int i = 1;cycle.empty() && i <= N;i++)
+		dfs(i);
+	if(cycle.empty())
+		printf("No cycle found!\n");
+	else
+	{
+		reverse(cycle.begin(), cycle.end());
+		printf("Cycle of length %u found!\n", cycle.size());
+		for(int n : cycle) printf("%d ", n);
+		printf("\n");
+	}
+}
+```
 
 ### Problems
 
@@ -149,5 +197,3 @@ For non-functional directed graphs, this code will still detect a cycle if it ex
  - [CSES Round Trip (undirected)](https://cses.fi/problemset/task/1669)
  - [CSES Round Trip II (directed)](https://cses.fi/problemset/task/1678)
 
-
-Cycle finding is also related to **strongly connected components**, a platinum level concept.