CGAL 5.6.3 - CGAL and the Boost Graph Library

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Specializations of boost::graph_traits
CGAL and the Boost Graph Library Reference

##

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`, `boost::edge_list_graph_tag`, and `boost::adjacency_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`, `boost::edge_list_graph_tag`, and`boost::adjacency_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`, `boost::edge_list_graph_tag`, and `boost::adjacency_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 functions:
  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` | `Vertex_around_target_iterator<Arrangement_2>` | An iterator to traverse through the vertices adjacent to a vertex. Its value type is `vertex_descriptor`.
`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`, `boost::edge_list_graph_tag`, and `boost::adjacency_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<K>::VertexHandle` | Identify vertices in the graph.
`edge_descriptor` | `unspecified_type` | Identify edges in the graph.
`halfedge_descriptor` | `OpenMesh::PolyMesh_ArrayKernelT<K>::HalfedgeHandle` | Identify halfedges in the graph.
`face_descriptor` | `OpenMesh::PolyMesh_ArrayKernelT<K>::FaceHandle` | Identify faces in the graph.
`adjacency_iterator` | `CGAL::Vertex_around_target_iterator<OpenMesh::PolyMesh_ArrayKernelT<K> >` | 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<K>::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<K>::HalfedgeIter` | An iterator to traverse through the halfedges of the graph. Its value type is `halfedge_descriptor`.
`face_iterator` | `OpenMesh::PolyMesh_ArrayKernelT<K>::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`,`boost::edge_list_graph_tag`, and`boost::adjacency_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<K>::VertexHandle` | Identify vertices in the graph.
`edge_descriptor` | `unspecified_type` | Identify edges in the graph.
`halfedge_descriptor` | `OpenMesh::TriMesh_ArrayKernelT<K>::HalfedgeHandle` | Identify halfedges in the graph.
`face_descriptor` | `OpenMesh::TriMesh_ArrayKernelT<K>::FaceHandle` | Identify faces in the graph.
`adjacency_iterator` | `CGAL::Vertex_around_target_iterator<OpenMesh::TriMesh_ArrayKernelT<K> >` | 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<K>::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<K>::HalfedgeIter` | An iterator to traverse through the halfedges of the graph. Its value type is `halfedge_descriptor`.
`face_iterator` | `OpenMesh::TriMesh_ArrayKernelT<K>::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`, `boost::edge_list_graph_tag`, and `boost::adjacency_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.