/**
  * Add a vertex to reach.
  *
  * @param k vertex' label
  * @param v the vertex
  * @param r the reach structure.
  */
 private void addToReach(Integer k, V v, HashMap<Integer, HashSet<V>> r) {
   if (r.containsKey(k)) {
     r.get(k).add(v);
   } else {
     HashSet<V> set = new HashSet<>();
     set.add(v);
     r.put(k, set);
   }
 }
  /**
   * Compute the minimal triangulation of the graph. Implementation of Algorithm MCS-M+ as described
   * in Berry et al. (2010), DOI:10.3390/a3020197 <a href="http://www.mdpi.com/1999-4893/3/2/197">
   * http://www.mdpi.com/1999-4893/3/2/197</a>
   */
  private void computeMinimalTriangulation() {
    // initialize chordGraph with same vertices as graph
    chordalGraph = new SimpleGraph<>(graph.getEdgeFactory());
    for (V v : graph.vertexSet()) {
      chordalGraph.addVertex(v);
    }

    // initialize g' as subgraph of graph (same vertices and edges)
    final UndirectedGraph<V, E> gprime = copyAsSimpleGraph(graph);
    int s = -1;
    generators = new ArrayList<>();
    meo = new LinkedList<>();

    final Map<V, Integer> vertexLabels = new HashMap<>();
    for (V v : gprime.vertexSet()) {
      vertexLabels.put(v, 0);
    }
    for (int i = 1, n = graph.vertexSet().size(); i <= n; i++) {
      V v = getMaxLabelVertex(vertexLabels);
      LinkedList<V> Y = new LinkedList<>(Graphs.neighborListOf(gprime, v));

      if (vertexLabels.get(v) <= s) {
        generators.add(v);
      }

      s = vertexLabels.get(v);

      // Mark x reached and all other vertices of gprime unreached
      HashSet<V> reached = new HashSet<>();
      reached.add(v);

      // mark neighborhood of x reached and add to reach(label(y))
      HashMap<Integer, HashSet<V>> reach = new HashMap<>();

      // mark y reached and add y to reach
      for (V y : Y) {
        reached.add(y);
        addToReach(vertexLabels.get(y), y, reach);
      }

      for (int j = 0; j < graph.vertexSet().size(); j++) {
        if (!reach.containsKey(j)) {
          continue;
        }
        while (reach.get(j).size() > 0) {
          // remove a vertex y from reach(j)
          V y = reach.get(j).iterator().next();
          reach.get(j).remove(y);

          for (V z : Graphs.neighborListOf(gprime, y)) {
            if (!reached.contains(z)) {
              reached.add(z);
              if (vertexLabels.get(z) > j) {
                Y.add(z);
                E fillEdge = graph.getEdgeFactory().createEdge(v, z);
                fillEdges.add(fillEdge);
                addToReach(vertexLabels.get(z), z, reach);
              } else {
                addToReach(j, z, reach);
              }
            }
          }
        }
      }

      for (V y : Y) {
        chordalGraph.addEdge(v, y);
        vertexLabels.put(y, vertexLabels.get(y) + 1);
      }

      meo.addLast(v);
      gprime.removeVertex(v);
      vertexLabels.remove(v);
    }
  }