What are faces of two triangles by planar_face_traversal?

[fililili]

Suppose we have two triangles, one have vertexes 0, 1, 2, the other have vertexes 3, 4, 5, then we use planar_face_traversal to traversal it, we will get 4 faces: New face: 0 1 2 New face: 1 0 2 New face: 3 4 5 New face: 4 3 5 But I think we should only have three faces, we only have three regions in fact. Anybody can explain the result? Here is the code (https://godbolt.org/z/eKKffh1Mx):

#include <iostream>
#include <boost/graph/adjacency_list.hpp>
#include <boost/graph/properties.hpp>
#include <boost/graph/graph_traits.hpp>
#include <boost/property_map/property_map.hpp>
#include <boost/ref.hpp>
#include <vector>

#include <boost/graph/planar_face_traversal.hpp>
#include <boost/graph/boyer_myrvold_planar_test.hpp>

using namespace boost;

// Some planar face traversal visitors that will
// print the vertices and edges on the faces

struct output_visitor : public planar_face_traversal_visitor
{
    void begin_face() { std::cout << "New face: "; }
    void end_face() { std::cout << std::endl; }
};

struct vertex_output_visitor : public output_visitor
{
    template < typename Vertex > void next_vertex(Vertex v)
    {
        std::cout << v << " ";
    }
};

struct edge_output_visitor : public output_visitor
{
    template < typename Edge > void next_edge(Edge e) { std::cout << e << " "; }
};

int main(int argc, char** argv)
{

    typedef adjacency_list< vecS, vecS, undirectedS,
        property< vertex_index_t, int >, property< edge_index_t, int > >
        graph;

    // Create a graph - this is a biconnected, 3 x 3 grid.
    // It should have four small (four vertex/four edge) faces and
    // one large face that contains all but the interior vertex
    graph g(6);

    add_edge(0, 1, g);
    add_edge(1, 2, g);
    add_edge(2, 0, g);

    add_edge(3, 4, g);
    add_edge(4, 5, g);
    add_edge(5, 3, g);

    // Initialize the interior edge index
    property_map< graph, edge_index_t >::type e_index = get(edge_index, g);
    graph_traits< graph >::edges_size_type edge_count = 0;
    graph_traits< graph >::edge_iterator ei, ei_end;
    for (auto [ei, ei_end] = edges(g); ei != ei_end; ++ei)
        put(e_index, *ei, edge_count++);

    // Test for planarity - we know it is planar, we just want to
    // compute the planar embedding as a side-effect
    typedef std::vector< graph_traits< graph >::edge_descriptor > vec_t;
    std::vector< vec_t > embedding(num_vertices(g));
    if (boyer_myrvold_planarity_test(boyer_myrvold_params::graph = g,
            boyer_myrvold_params::embedding = &embedding[0]))
        std::cout << "Input graph is planar" << std::endl;
    else
        std::cout << "Input graph is not planar" << std::endl;

    std::cout << std::endl << "Vertices on the faces: " << std::endl;
    vertex_output_visitor v_vis;
    planar_face_traversal(g, &embedding[0], v_vis);

    std::cout << std::endl << "Edges on the faces: " << std::endl;
    edge_output_visitor e_vis;
    planar_face_traversal(g, &embedding[0], e_vis);

    return 0;
}

[fililili]

If I add one more edge 0 to 3 ( add_edge(3, 0, g); ), we only have three faces. Add one edge will reduce the number of faces. Here is the code (https://godbolt.org/z/f8dGbeY43) I think the algorithm can only handle connective graph.