For the most part, the differences between member functions and free functions are syntactic, and not very important, though people can get religious about them. However, we had one technical reason for favoring free functions. A programmer can overload a free function for a type coming from a 3rd party without modifying the source code/definition of that type. There are several uses of this in the BGL. For example, Stanford GraphBase and LEDA graphs can both be used in BGL algorithms because of overloads in stanford_graph.hpp and leda_graph.hpp. One can even use std::vector<std::list> as a graph due to the overloads in vector_as_graph.hpp.
Of course, there is a way to adapt 3rd party classes into an interface with member functions. You create an adaptor class. However, the disadvantage of an adaptor class (compared to overloaded functions) is that one has to physically wrap and unwrap the objects as they go into/out of BGL algorithms. So the overloaded function route is more convenient. Granted, this is not a huge difference, but since there weren't other strong reasons, it was enough for us to choose member functions.
Our religious reason for choosing free functions is to send the message that BGL is a generic library, and not a traditional object-oriented library. OO was hip in the 80s and 90s, but its time we moved beyond!
The example ordered_out_edges.cpp shows how to do this.
// specify the graph type typedef adjacency_list<listS, listS, undirectedS, property<vertex_index_t, std::size_t>, no_property > graph_t; // construct a graph object graph_t G(num_nodes); // obtain a property map for the vertex_index property property_map<graph_t, vertex_index_t>::type index = get(vertex_index, G); // initialize the vertex_index property values graph_traits<graph_t>::vertex_iterator vi, vend; graph_traits<graph_t>::vertices_size_type cnt = 0; for(tie(vi,vend) = vertices(G); vi != vend; ++vi) put(index, *vi, cnt++);
../../../boost/concept_check.hpp:209: no match for `boost::detail::error_property_not_found & == boost::detail::error_property_not_found &'or a message such as:
../../..\boost/graph/depth_first_search.hpp(78) : error C2664: 'white' : cannot convert parameter 1 from 'struct boost::detail::error_property_not_found' to 'enum boost::default_color_type'The reason is that the algorithm expected to find some property (like color or weight) attached to the vertices or edges of the graph, but didn't find it. You need to either add an interior property to the graph, or create an exterior property map for the property and pass it as an argument to the algorithm.