// Uses depth-first search to check if the graph induced by the subgraph given as a parameter is connected
// In other words, we retreive all edges from the original graph and check the subgraph for connectedness
public static bool Connected(Graph graph, BitSet subgraph)
{
// Vertices that are visited
Set<int> visited = new Set<int>();
// Stack of vertices yet to visit
Stack<int> stack = new Stack<int>();
// Initial vertex
int s = subgraph.First();
stack.Push(s);
// Continue while there are vertices on the stack
while (stack.Count > 0)
{
int v = stack.Pop();
// If we have not encountered this vertex before, then we check for all neighbors if they are part of the subgraph
// If a neighbor is part of the subgraph it means that we have to push it on the stack to explore it at a later stage
if (!visited.Contains(v))
{
visited.Add(v);
foreach (int w in graph.OpenNeighborhood(v))
if (subgraph.Contains(w))
stack.Push(w);
}
}
// If we visited an equal number of vertices as there are vertices in the subgraph then the subgraph is connected
return visited.Count == subgraph.Count;
}
// Returns all connected components in a certain subgraph, where we define a subgraph by the vertices that are contained in it
// We apply multiple dfs searches to find all connected parts of the graph
public static List<BitSet> ConnectedComponents(Graph graph, BitSet subgraph)
{
// Each connected component is defined as a bitset, thus the result is a list of these bitsets
List<BitSet> result = new List<BitSet>();
// Working stack for the dfs algorithm
Stack<int> stack = new Stack<int>();
// Each vertex of the subgraph will either be explored, or it will be the starting point of a new dfs search
BitSet todo = subgraph.Copy();
while (!todo.IsEmpty)
{
int s = todo.First();
stack.Push(s);
// Start tracking the new component
BitSet component = new BitSet(0, graph.Size);
// Default dfs exploring part of the graph
while (stack.Count > 0)
{
int v = stack.Pop();
// If we have not encountered this vertex before (meaning it isn't in this component), then we check for all neighbors if they are part of the subgraph
// If a neighbor is part of the subgraph it means that we have to push it on the stack to explore it at a later stage for this component
if (!component.Contains(v))
{
component.Add(v);
// Remove this vertex from the 'todo' list, since it can never be the starting point of a new component
todo.Remove(v);
foreach (int w in graph.OpenNeighborhood(v))
if (subgraph.Contains(w))
stack.Push(w);
}
}
// The whole connected component has been found, so we can add it to the list of results
result.Add(component);
}
return result;
}
