node  ST_NUMBERING(const graph& G, node_array<int>& stnum, list<node>& stlist, edge e_st = nil)  
ST_NUMBERING computes an stnumbering of G. If e_st is nil then
t is set to some arbitrary node of G. The node s is set to a
neighbor of t and is returned. If e_st is not nil then s is set
to the source of e_st and t is set to its target.
The nodes of G are numbered such
that s has number 1, t has number n, and every node v different
from s and t has a smaller and a larger numbered neighbor.
The ordered list of nodes is returned in stlist. If G has no nodes
then nil is returned and if G has exactly one node then this node is returned and given number one. Precondition G is biconnected. 

bool  PLANAR(graph& , bool embed=false)  
PLANAR takes as input a directed graph G(V, E) and performs a planarity test
for it. G must not contain selfloops. If the second argument embed has
value true and G is a planar
graph it is transformed into a planar map (a combinatorial embedding such that
the edges in all adjacency lists are in clockwise ordering). PLANAR returns
true if G is planar and false otherwise.
The algorithm ([45]) has running time
O( V +  E).


bool  PLANAR(graph& G, list<edge>& el, bool embed=false)  
PLANAR takes as input a directed graph G(V, E) and performs a planarity test
for G. PLANAR returns true if G is planar and false otherwise.
If G is not planar a KuratowskySubgraph is computed and returned in el.


bool  CHECK_KURATOWSKI(const graph& G, const list<edge>& el)  
returns true if all edges in el are edges of G and if the edges in el form a Kuratowski subgraph of G, returns false otherwise. Writes diagnostic output to cerr.


int  KURATOWSKI(graph& G, list<node>& V, list<edge>& E, node_array<int>& deg)  
KURATOWKI computes a Kuratowski subdivision K of G as follows. V is the
list of all nodes and subdivision points of K. For all v V the degree
deg[v] is equal to 2 for subdivision points, 4 for all other nodes if K
is a K_{5}, and 3 (+3) for the nodes of the left (right) side if K is a
K_{3, 3}. E is the list of all edges in the Kuratowski subdivision.


list<edge>  TRIANGULATE_PLANAR_MAP(graph& G)  
TRIANGULATE_PLANAR_MAP takes a directed graph G representing a planar map.
It triangulates the faces of G by inserting additional edges. The list of
inserted edges is returned.
Precondition G must be connected. The algorithm ([47]) has running time O( V +  E).


void  FIVE_COLOR(graph& G, node_array<int>& C)  
colors the nodes of G using 5 colors, more precisely, computes for every node v a color C[v] {0,..., 4}, such that C[source(e)]! = C[target(e)] for every edge e. Precondition G is planar. Remark: works also for many (sparse ?) nonplanar graph.  
void  INDEPENDENT_SET(const graph& G, list<node>& I)  
determines an independent set of nodes I in G. Every node in I has degree at most 9. If G is planar and has no parallel edges then I contains at least n/6 nodes.  
bool  Is_CCW_Ordered(const graph& G, const node_array<int>& x, const node_array<int>& y)  
checks whether the cyclic adjacency list of any node v agrees with the counterclockwise ordering of the neighbors of v around v defined by their geometric positions.  
bool  SORT_EDGES(graph& G, const node_array<int>& x, const node_array<int>& y)  
reorders all adjacency lists such the cyclic adjacency list of any node v agrees with the counterclockwise order of v's neighbors around v defined by their geometric positions. The function returns true if G is a plane map after the call.  
bool  Is_CCW_Ordered(const graph& G, const edge_array<int>& dx, const edge_array<int>& dy)  
checks whether the cyclic adjacency list of any node v agrees with the counterclockwise ordering of the neighbors of v around v. The direction of edge e is given by the vector (dx(e), dy(e)).  
bool  SORT_EDGES(graph& G, const edge_array<int>& dx, const edge_array<int>& dy)  
reorders all adjacency lists such the cyclic adjacency list of any node v agrees with the counterclockwise order of v's neighbors around v. The direction of edge e is given by the vector (dx(e), dy(e)). The function returns true if G is a plane map after the call. 