The BGL defines the class template boost::graph_traits as a uniform interface to the properties and types of graph types.

We provide specializations of this class template for several CGAL data structures.

Specialization for the Surface_mesh Class

Defined in <CGAL/boost/graph/graph_traits_Surface_mesh.h>

We provide partial specialization for the class CGAL::Surface_mesh so that it is a model of the graph concepts BidirectionalGraph, VertexAndEdgeListGraph, AdjacencyMatrix, and MutableFaceGraph.

The const specialization, boost::graph_traits< CGAL::Surface_mesh<Traits> const> is also defined, using the constant handles in the surface mesh.

The traits class boost::graph_traits< CGAL::Surface_mesh<T> > provides the following types:

Member	Value	Description
`vertex_descriptor`	`Surface_mesh::Vertex_index`	Identify vertices in the graph.
`edge_descriptor`	`Surface_mesh::Edge_index`	Identify edges in the graph.
`halfedge_descriptor`	`Surface_mesh::Halfedge_index`	Identify halfedges in the graph.
`face_descriptor`	`Surface_mesh::Face_index`	Identify faces in the graph.
`adjacency_iterator`	`CGAL::Vertex_around_target_iterator<Surface_mesh<P> >`	An iterator to traverse through the vertices adjacent to a vertex. Its value type is `vertex_descriptor`.
`out_edge_iterator`	`CGAL::Out_edge_iterator<Surface_mesh<P> >`	An iterator to traverse through the outgoing edges incident to a vertex. Its value type is `edge_descriptor`.
`in_edge_iterator`	`CGAL::In_edge_iterator<Surface_mesh<P> >`	An iterator to traverse through the incoming edges incident to a vertex. Its value type is `edge_descriptor`.
`vertex_iterator`	`Surface_mesh::Vertex_iterator`	An iterator to traverse through the vertices of the graph. Its value type is `vertex_descriptor`.
`edge_iterator`	`Surface_mesh::Edge_iterator`	An iterator to traverse through the edges of the graph. Its value type is `edge_descriptor`.
`halfedge_iterator`	`Surface_mesh::Halfedge_iterator`	An iterator to traverse through the halfedges of the graph. Its value type is `halfedge_descriptor`.
`face_iterator`	`Surface_mesh::Face_iterator`	An iterator to traverse through the faces of the graph. Its value type is `face_descriptor`.
`directed_category`	`boost::undirected_tag`	This graph is not directed.
`edge_parallel_category`	`boost::disallow_parallel_edge_tag`	This graph does not support multiedges.
`traversal_category`	Inherits from `boost::bidirectional_graph_tag`, `boost::vertex_list_graph_tag`, and `boost::edge_list_graph_tag`	The ways in which the vertices in the graph can be traversed.
`vertices_size_type`	`Surface_mesh::vertices_size_type`	The size type of the vertex list.
`edges_size_type`	`Surface_mesh::edges_size_type`	The size type of the edge list.
`degree_size_type`	`Surface_mesh::degree_size_type`	The size type of the adjacency list.

Specialization for the Polyhedron Class

Defined in <CGAL/boost/graph/graph_traits_Polyhedron_3.h>

We provide partial specialization for the class CGAL::Polyhedron_3 so that it is a model of the graph concepts BidirectionalGraph, VertexAndEdgeListGraph, AdjacencyMatrix, and MutableFaceGraph.

The const specialization, boost::graph_traits< CGAL::Polyhedron_3<Traits> const> is also defined, using the constant handles in the polyhedron.

The traits class boost::graph_traits< CGAL::Polyhedron_3<T> > provides the following types:

Member	Value	Description
`vertex_descriptor`	`Polyhedron_3::Vertex_handle`	Identify vertices in the graph.
`edge_descriptor`	`unspecified_type`	Identify edges in the graph.
`halfedge_descriptor`	`Polyhedron_3::Halfedge_handle`	Identify halfedges in the graph.
`face_descriptor`	`Polyhedron_3::Face_handle`	Identify faces in the graph.
`adjacency_iterator`	`CGAL::Vertex_around_target_iterator<Polyhedron_3<T> >`	An iterator to traverse through the vertices adjacent to a vertex. Its value type is `vertex_descriptor`.
`out_edge_iterator`	`CGAL::Out_edge_iterator<Polyhedron_3<T> >`	An iterator to traverse through the outgoing edges incident to a vertex. Its value type is `edge_descriptor`.
`in_edge_iterator`	`CGAL::In_edge_iterator<Polyhedron_3<T> >`	An iterator to traverse through the incoming edges incident to a vertex. Its value type is `edge_descriptor`.
`vertex_iterator`	`unspecified_type`	An iterator to traverse through the vertices of the graph. Its value type is `vertex_descriptor`.
`edge_iterator`	`unspecified_type`	An iterator to traverse through the edges of the graph. Its value type is `edge_descriptor`.
`halfedge_iterator`	`unspecified_type`	An iterator to traverse through the halfedges of the graph. Its value type is `halfedge_descriptor`.
`face_iterator`	`unspecified_type`	An iterator to traverse through the faces of the graph. Its value type is `face_descriptor`.
`directed_category`	`boost::undirected_tag`	This graph is not directed.
`edge_parallel_category`	`boost::disallow_parallel_edge_tag`	This graph does not support multiedges.
`traversal_category`	Inherits from `boost::bidirectional_graph_tag`, `boost::vertex_list_graph_tag`, and `boost::edge_list_graph_tag`	The ways in which the vertices in the graph can be traversed.
`vertices_size_type`	`Polyhedron_3::size_type`	The size type of the vertex list.
`edges_size_type`	`Polyhedron_3::size_type`	The size type of the edge list.
`degree_size_type`	`Polyhedron_3::size_type`	The size type of the adjacency list.

For convenience, the type edge_descriptor is hashable using the functor CGAL::Handle_hash_function, which is the default hash functor of CGAL::Unique_hash_map.

Specialization for the Linear_cell_complex_for_combinatorial_map Class

Defined in <CGAL/boost/graph/graph_traits_Linear_cell_complex_for_combinatorial_map.h>

We provide partial specialization for the class CGAL::Linear_cell_complex_for_combinatorial_map so that it is a model of the graph concepts BidirectionalGraph, VertexAndEdgeListGraph, AdjacencyMatrix, and MutableFaceGraph.

The const specialization is also defined, using the constant handles in the linear cell complex.

Let us denote by LCC an instantiation of CGAL::Linear_cell_complex_for_combinatorial_map<...> class. The traits class boost::graph_traits<LCC> provides the following types:

Member	Value	Description
`vertex_descriptor`	`LCC::Vertex_attribute_handle`	The vertex descriptor
`edge_descriptor`	`unspecified_type`	The edge descriptor
`halfedge_descriptor`	`LCC::Dart_handle`	The halfedge descriptor
`face_descriptor`	`unspecified_type`	The face descriptor
`adjacency_iterator`	`CGAL::Vertex_around_target_iterator<LCC>`	Iterates through adjacent vertices
`out_edge_iterator`	`CGAL::Out_edge_iterator<LCC>`	Iterate through the out-edges of a vertex.
`in_edge_iterator`	`CGAL::Out_edge_iterator<LCC>`	Iterate through the in-edges of a vertex.
`vertex_iterator`	`unspecified_type`	Iterate through the vertices of LCC.
`edge_iterator`	`unspecified_type`	Iterate through the edges of LCC.
`halfedge_iterator`	`unspecified_type`	Iterate through the halfedges of LCC.
`face_iterator`	`unspecified_type`	Iterate through the faces of LCC.
`directed_category`	Inherits from `boost::bidirectional_graph_tag`, `boost::vertex_list_graph_tag` and `boost::edge_list_graph_tag`
`edge_parallel_category`	`boost::disallow_parallel_edge_tag`	Indicates that this graph does not support multiedges
`traversal_category`	`boost::bidirectional_graph_tag`	Indicates that this graph is bidirectional
`vertices_size_type`	`LCC::size_type`	The size type of the vertex list
`edges_size_type`	`LCC::size_type`	The size type of the edge list
`degree_size_type`	`LCC::size_type`	The size type of the adjacency list

For convenience, the type edge_descriptor is hashable using the functor CGAL::Handle_hash_function that is the default hash functor of CGAL::Unique_hash_map.

Requirements

The item class used by CGAL::Linear_cell_complex_for_combinatorial_map must have both 0-attributes and 2-attributes enabled.
No dart is 1-free, nor 2-free. Holes in a mesh are represented by using the same convention than for CGAL::Polyhedron_3 and CGAL::Surface_mesh: a dart d belongs to a border if the 2-attribute of beta<2>(d) is nullptr.
All darts of the linear cell complexes must be associated with a 2-attribute, except darts that represent holes.
In order to use property maps, darts and types associated with of 0- and 2-attributes must define the two fonctions:
int id() const; // Returns the index.
int& id(); // Returns a reference to the index stored in the attribute.

For darts, this can be done by defining Darts_with_id as CGAL::Tag_true in the Dart_wrapper struct of the item class.
For attributes, it is possible to use CGAL::Cell_attribute_with_id and CGAL::Cell_attribute_with_point_and_id classes to define your item class using attributes with id.

You can also use the CGAL::Linear_cell_complex_bgl_min_items item class, or you can use directly the CGAL::Linear_cell_complex_for_bgl_combinatorial_map_helper class.

Specializations for the 2D Triangulation Classes

Defined in <CGAL/boost/graph/graph_traits_Triangulation_2.h>, <CGAL/boost/graph/graph_traits_Delaunay_triangulation_2.h>, <CGAL/boost/graph/graph_traits_Regular_triangulation_2.h>, <CGAL/boost/graph/graph_traits_Constrained_Delaunay_triangulation_2.h>, <CGAL/boost/graph/graph_traits_Constrained_triangulation_2.h>, <CGAL/boost/graph/graph_traits_Constrained_triangulation_plus_2.h>, and <CGAL/boost/graph/graph_traits_Triangulation_hierarchy_2.h>.

We provide partial specialization for the classes CGAL::Triangulation_2, CGAL::Delaunay_triangulation_2, CGAL::Regular_triangulation_2, CGAL::Constrained_triangulation_2, CGAL::Constrained_Delaunay_triangulation_2, CGAL::Constrained_triangulation_plus_2, and CGAL::Triangulation_hierarchy_2 so that they are model of the graph concepts BidirectionalGraph, VertexAndEdgeListGraph, and FaceListGraph.

Only finite simplices exist when viewed through the scope of these graph traits classes. The infinite vertex, halfedges, edges, and faces will thus not appear when looping around a border vertex, or walking through the faces container.

The mapping between vertices, edges, and faces of the triangulation and the graph is rather straightforward, but there are some subtleties. The value type of the BGL iterators is the vertex or edge descriptor, whereas in CGAL all iterators and circulators are also handles and hence have as value type Vertex or Edge.

Member	Value	Description
`vertex_descriptor`	`unspecified_type`	Identify vertices in the graph.
`halfedge_descriptor`	`unspecified_type`	Identify halfedges in the graph. It is constructible from and convertible to `Triangulation::Edge`. It is not a simple typedef, but a proper class, because there is no representation for halfedges in the 2D triangulation data structure.
`edge_descriptor`	`unspecified_type`	Identify edges in the graph. It is constructible from and convertible to `Triangulation::Edge`. It is not a simple typedef, but a proper class, because in an undirected graph the edges `(u,v)` and `(v,u)` must be equal. This is not the case for the Edge type of the triangulation.
`face_descriptor`	`unspecified_type`	Identify faces in the graph.
`adjacency_iterator`	`unspecified_type`	An iterator to traverse through the vertices adjacent to a vertex. Its value type is `vertex_descriptor`.
`out_edge_iterator`	`unspecified_type`	An iterator to traverse through the outgoing edges incident to a vertex. Its value type is `edge_descriptor`.
`in_edge_iterator`	`unspecified_type`	An iterator to traverse through the incoming edges incident to a vertex. Its value type is `edge_descriptor`.
`vertex_iterator`	`unspecified_type`	An iterator to traverse through the vertices of the graph. Its value type is `vertex_descriptor`.
`halfedge_iterator`	`unspecified_type`	An iterator to traverse through the halfedges of the graph. Its value type is `halfedge_descriptor`.
`edge_iterator`	`unspecified_type`	An iterator to traverse through the edges of the graph. Its value type is `edge_descriptor`.
`face_iterator`	`unspecified_type`	An iterator to traverse through the faces of the graph. Its value type is `face_descriptor`.
`directed_category`	`boost::undirected_tag`	This graph is not directed.
`edge_parallel_category`	`boost::disallow_parallel_edge_tag`	This graph does not support multiedges.
`traversal_category`	Inherits from `boost::bidirectional_graph_tag`, `boost::adjacency_graph_tag`, `boost::vertex_list_graph_tag`, and `boost::edge_list_graph_tag`	The ways in which the vertices in the graph can be traversed..
`vertices_size_type`	`Triangulation::size_type`	The size type of the vertex list.
`edges_size_type`	`Triangulation::size_type`	The size type of the edge list.
`degree_size_type`	`Triangulation::size_type`	The size type of the adjacency list.

Specialization for the Arrangement Classes

Defined in <CGAL/boost/graph/graph_traits_Arrangement_2.h>

We provide partial specialization for the class CGAL::Arrangement_2 so that it is model of the graph concepts BidirectionalGraph and VertexAndEdgeListGraph.

The const specialization, boost::graph_traits< CGAL::Arrangement_2<Traits, Dcel> const> is also defined, using the constant handles in the arrangement.

The traits class boost::graph_traits< CGAL::Arrangement_2<T, Dcel> > provides the following types:

Member	Value	Description
`vertex_descriptor`	`Arrangement_2::Vertex_handle`	Identify vertices in the graph.
`edge_descriptor`	`Arrangement_2::Halfedge_handle`	Identify edges in the graph.
`adjacency_iterator`	Not provided
`out_edge_iterator`	`unspecified_type`	An edge iterator which only iterates over the outgoing halfedges around a vertex. It corresponds to a `Arrangement_2::Halfedge_around_vertex_circulator` with the difference that its value type is an edge descriptor and not `Arrangement_2::Halfedge`
`in_edge_iterator`	`unspecified_type`	An edge iterator which only iterates over the incoming edges around a vertex. It corresponds to a `Arrangement_2::Halfedge_around_vertex_circulator` with the difference that its value type is an edge descriptor and not `Arrangement_2::Halfedge`
`vertex_iterator`	`unspecified_type`	An iterator corresponding to `Arrangement_2::Vertex_iterator`, with the difference that its value type is a vertex descriptor and not `Arrangement_2::Vertex`
`edge_iterator`	`unspecified_type`	An iterator corresponding to `Arrangement_2::Halfedge_iterator` with the difference that its value type is an edge descriptor and not `Arrangement_2::Halfedge`
`directed_category`	`boost::directed_tag`	This graph is directed.
`edge_parallel_category`	`boost::allow_parallel_edge_tag`	This graph supports multiedges.
`traversal_category`	Inherits from `boost::bidirectional_graph_tag`, `boost::vertex_list_graph_tag`, and `boost::edge_list_graph_tag`	The ways in which the vertices in the graph can be traversed.
`vertices_size_type`	`Arrangement_2::Size`	The size type of the vertex list.
`edges_size_type`	`Arrangement_2::Size`	The size type of the edge list.
`degree_size_type`	`Arrangement_2::Size`	The size type of the adjacency list.

Specialization for the OpenMesh::PolyMesh_ArrayKernelT Class

Defined in <CGAL/boost/graph/graph_traits_PolyMesh_ArrayKernelT.h>

We provide partial specialization for the class OpenMesh::PolyMesh_ArrayKernelT so that it is a model of the graph concepts BidirectionalGraph, VertexAndEdgeListGraph, AdjacencyMatrix, and MutableFaceGraph.

The traits class boost::graph_traits<OpenMesh::PolyMesh_ArrayKernelT<K> > provides the following types:

Member	Value	Description
`vertex_descriptor`	`OpenMesh::PolyMesh_ArrayKernelT::VertexHandle`	Identify vertices in the graph.
`edge_descriptor`	`unspecified_type`	Identify edges in the graph.
`halfedge_descriptor`	`OpenMesh::PolyMesh_ArrayKernelT::HalfedgeHandle`	Identify halfedges in the graph.
`face_descriptor`	`OpenMesh::PolyMesh_ArrayKernelT::FaceHandle`	Identify faces in the graph.
`adjacency_iterator`	`unspecified_type`	An iterator to traverse through the vertices adjacent to a vertex. Its value type is `vertex_descriptor`.
`out_edge_iterator`	`CGAL::Out_edge_iterator<OpenMesh::PolyMesh_ArrayKernelT<K> >`	An iterator to traverse through the outgoing edges incident to a vertex. Its value type is `edge_descriptor`.
`in_edge_iterator`	`CGAL::In_edge_iterator<OpenMesh::PolyMesh_ArrayKernelT<K> >`	An iterator to traverse through the incoming edges incident to a vertex. Its value type is `edge_descriptor`.
`vertex_iterator`	`OpenMesh::PolyMesh_ArrayKernelT::VertexIter`	An iterator to traverse through the vertices of the graph. Its value type is `vertex_descriptor`.
`edge_iterator`	`unspecified_type`	An iterator to traverse through the edges of the graph. Its value type is `edge_descriptor`.
`halfedge_iterator`	`OpenMesh::PolyMesh_ArrayKernelT::HalfedgeIter`	An iterator to traverse through the halfedges of the graph. Its value type is `halfedge_descriptor`.
`face_iterator`	`OpenMesh::PolyMesh_ArrayKernelT::FaceIter`	An iterator to traverse through the faces of the graph. Its value type is `face_descriptor`.
`directed_category`	`boost::undirected_tag`	This graph is not directed.
`edge_parallel_category`	`boost::disallow_parallel_edge_tag`	This graph does not support multiedges.
`traversal_category`	Inherits from `boost::bidirectional_graph_tag`, `boost::vertex_list_graph_tag`, and `boost::edge_list_graph_tag`	The ways in which the vertices in the graph can be traversed.
`vertices_size_type`	`unsigned int`	The size type of the vertex list.
`edges_size_type`	`unsigned int`	The size type of the edge list.
`degree_size_type`	`unsigned int`	The size type of the adjacency list.

Specialization for the OpenMesh::TriMesh_ArrayKernelT Class

Defined in <CGAL/boost/graph/graph_traits_TriMesh_ArrayKernelT.h>

We provide partial specialization for the class OpenMesh::TriMesh_ArrayKernelT so that it is a model of the graph concepts BidirectionalGraph, VertexAndEdgeListGraph, AdjacencyMatrix, and MutableFaceGraph.

The traits class boost::graph_traits<OpenMesh::TriMesh_ArrayKernelT<K> > provides the following types:

Member	Value	Description
`vertex_descriptor`	`OpenMesh::TriMesh_ArrayKernelT::VertexHandle`	Identify vertices in the graph.
`edge_descriptor`	`unspecified_type`	Identify edges in the graph.
`halfedge_descriptor`	`OpenMesh::TriMesh_ArrayKernelT::HalfedgeHandle`	Identify halfedges in the graph.
`face_descriptor`	`OpenMesh::TriMesh_ArrayKernelT::FaceHandle`	Identify faces in the graph.
`adjacency_iterator`	`unspecified_type`	An iterator to traverse through the vertices adjacent to a vertex. Its value type is `vertex_descriptor`.
`out_edge_iterator`	`CGAL::Out_edge_iterator<OpenMesh::TriMesh_ArrayKernelT<K> >`	An iterator to traverse through the outgoing edges incident to a vertex. Its value type is `edge_descriptor`.
`in_edge_iterator`	`CGAL::In_edge_iterator<OpenMesh::TriMesh_ArrayKernelT<K> >`	An iterator to traverse through the incoming edges incident to a vertex. Its value type is `edge_descriptor`.
`vertex_iterator`	`OpenMesh::PolyMesh_ArrayKernelT::VertexIter`	An iterator to traverse through the vertices of the graph. Its value type is `vertex_descriptor`.
`edge_iterator`	`unspecified_type`	An iterator to traverse through the edges of the graph. Its value type is `edge_descriptor`.
`halfedge_iterator`	`OpenMesh::TriMesh_ArrayKernelT::HalfedgeIter`	An iterator to traverse through the halfedges of the graph. Its value type is `halfedge_descriptor`.
`face_iterator`	`OpenMesh::TriMesh_ArrayKernelT::FaceIter`	An iterator to traverse through the faces of the graph. Its value type is `face_descriptor`.
`directed_category`	`boost::undirected_tag`	This graph is not directed.
`edge_parallel_category`	`boost::disallow_parallel_edge_tag`	This graph does not support multiedges.
`traversal_category`	Inherits from `boost::bidirectional_graph_tag`, `boost::vertex_list_graph_tag`, and `boost::edge_list_graph_tag`	The ways in which the vertices in the graph can be traversed.
`vertices_size_type`	`unsigned int`	The size type of the vertex list.
`edges_size_type`	`unsigned int`	The size type of the edge list.
`degree_size_type`	`unsigned int`	The size type of the adjacency list.

Specialization for the Seam_mesh Class

Defined in <CGAL/boost/graph/graph_traits_Seam_mesh.h>

We provide partial specialization for the class CGAL::Seam_mesh so that it is a model of the graph concepts BidirectionalGraph, VertexAndEdgeListGraph, AdjacencyMatrix, and FaceListGraph.

The traits class boost::graph_traits< CGAL::Seam_mesh<T> > provides the following types:

Member	Value	Description
`vertex_descriptor`	`Seam_mesh::vertex_descriptor`	Identify vertices in the graph.
`edge_descriptor`	`Seam_mesh::edge_descriptor`	Identify edges in the graph.
`halfedge_descriptor`	`Seam_mesh::halfedge_descriptor`	Identify halfedges in the graph.
`face_descriptor`	`Seam_mesh::face_descriptor`	Identify faces in the graph.
`adjacency_iterator`	`CGAL::Vertex_around_target_iterator<CGAL::Seam_mesh<T> >`	An iterator to traverse through the vertices adjacent to a vertex. Its value type is `vertex_descriptor`.
`out_edge_iterator`	`CGAL::Out_edge_iterator<CGAL::Seam_mesh<T> >`	An iterator to traverse through the outgoing edges incident to a vertex. Its value type is `edge_descriptor`.
`in_edge_iterator`	`CGAL::In_edge_iterator<CGAL::Seam_mesh<T> >`	An iterator to traverse through the incoming edges incident to a vertex. Its value type is `edge_descriptor`.
`vertex_iterator`	`Seam_mesh::vertex_iterator`	An iterator to traverse through the vertices of the graph. Its value type is `vertex_descriptor`.
`edge_iterator`	`Seam_mesh::edge_iterator`	An iterator to traverse through the edges of the graph. Its value type is `edge_descriptor`.
`halfedge_iterator`	`Seam_mesh::halfedge_iterator`	An iterator to traverse through the halfedges of the graph. Its value type is `halfedge_descriptor`.
`face_iterator`	`Seam_mesh::face_iterator`	An iterator to traverse through the faces of the graph. Its value type is `face_descriptor`.
`directed_category`	`boost::undirected_tag`	This graph is not directed.
`edge_parallel_category`	`boost::disallow_parallel_edge_tag`	This graph does not support multiedges.
`traversal_category`	Inherits from `boost::bidirectional_graph_tag`, `boost::vertex_list_graph_tag`, and `boost::edge_list_graph_tag`	The ways in which the vertices in the graph can be traversed.
`vertices_size_type`	`Seam_mesh::vertices_size_type`	The size type of the vertex list.
`edges_size_type`	`Seam_mesh::edges_size_type`	The size type of the edge list.
`degree_size_type`	`Seam_mesh::degree_size_type`	The size type of the adjacency list.