Schnyder woods for higher genus triangulated surfaces with ...

Schnyder woods for higher genus triangulated 

surfaces with applications to compression 

(a) 

v n−1 

(b) 

v n−1 

(c) 

v 0 v 1 

v 0 v 1 

Luca Castelli Aleardi, Eric Fusy and 

Thomas Lewiner 

MAT. 18/08

Cadastro de Pré–publicação : MAT. 18/08 

Título : 

Autores : 

Schnyder woods for higher genus triangulated surfaces with applications 

to compression 

Luca Castelli Aleardi, Eric Fusy and Thomas Lewiner 

Palavras Chave : 1. Schnyder woods 2. Schnyder trees 

3. Graph realizer 4. Handlebody theory 

5. High genus surface 6. Topological graph theory 

Número de páginas : 14 

Data de submissão : September 1 st , 2008 

Endereço eletrônico : 

//www.matmidia.mat.puc-rio.br/tomlew/attachments/realizer_dcg.pdf 

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Área de Concentração : □ Matemática Pura Matemática Aplicada 

Âmbito : □ Nacional Internacional 

Sem vínculado com tese. 

Linha de Pesquisa do Artigo : 

Projeto de Pesquisa (conforme relatório CAPES) : 

Projeto de Pesquisa vinculado (orgão financiador) : 

Computação Gráfica 

Compressão de malhas 

Edital CNPq Universal 

Outras Informações Relevantes : 

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Breve Descrição do Projeto (em case de projeto novo) : 

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Schnyder woods for higher genus triangulated surfaces with applications to compression 

LUCA CASTELLI ALEARDI 1 , ERIC FUSY 2 AND THOMAS LEWINER 3 

1 Department of Computer Science — Université Libre de Bruxelles — Bruxelles — Belgium 

2 Department of Mathematics — Simon Fraser University — Burnaby — Canada 

3 Department of Mathematics — Pontifícia Universidade Católica — Rio de Janeiro — Brazil 

http://www.lix.polytechnique.fr/˜{amturing,fusy}. http://www.matmidia.mat.puc-rio.br/tomlew. 

Abstract. Schnyder woods are a well known combinatorial structure for planar graphs, which yields a decomposition 

into 3 vertex-spanning trees. Our goal is to extend definitions and algorithms for Schnyder woods designed 

for planar graphs (corresponding to combinatorial surfaces with the topology of the sphere, i.e., of genus 0) to 

the more general case of graphs embedded on surfaces of arbitrary genus. First, we define a new traversal order 

of the vertices of a triangulated surface of genus g together with an orientation and coloration of the edges that 

extends the one proposed by Schnyder for the planar case. As a by-product we show how some recent schemes for 

compression and compact encoding of graphs can be extended to higher genus. All the algorithms presented here 

have linear time complexity. 

Keywords: Schnyder woods. Schnyder trees. Graph realizer. Handlebody theory. High genus surface. 

Topological graph theory. 

(a) 

v n−1 

(b) 

v n−1 

(c) 

v 0 v 1 

v 0 v 1 

Figure 1: (a) A rooted planar triangulation, (b) endowed with a Schnyder wood. (c) The local condition of Schnyder woods. 

1 Introduction 

Schnyder woods are a nice and deep combinatorial structure 

to finely capture the notion of planarity of a graph. They 

are named after W. Schnyder, who introduced these structures 

under the name of realizers and derived as main applications 

a new planarity criterion in terms of poset dimensions 

[32], as well as a very elegant and simple straightline 

drawing algorithm [33]. There are several equivalent 

formulations of Schnyder woods, either in terms of angle 

labeling (Schnyder labeling) or edge coloring and orientation 

or in terms of orientations with prescribed outdegrees. 

The most classical formulation is for the family of maximal 

plane graphs, i.e., plane triangulations, yielding the following 

striking property: the internal edges of a triangulation 

can be partitioned into three trees that span all inner 

vertices and are rooted respectively at each of the three 

Preprint MAT. 18/08, communicated on September 1 st , 2008 to the Department 

of Mathematics, Pontifícia Universidade Católica — Rio de Janeiro, 

Brazil. 

vertices incident to the outer face. Schnyder woods, and 

more generally α-orientations, received a great deal of attention 

[33, 15, 20, 18]. From the combinatorial point of 

view, the set of Schnyder woods of a fixed triangulation 

has an interesting lattice structure [7, 3, 16, 13, 28], and 

the nice characterization in terms of spanning trees motivated 

a large number of applications in several domains 

such as graph drawing [33, 20], graph coding and sampling 

[12, 19, 4, 29, 17, 6, 10, 2]. Previous work focused 

mainly on the application and extension of the combinatorial 

properties of Schnyder woods to 3-connected plane 

graphs [15, 20]. In this article, we deal with combinatorial 

surfaces possibly having handles, i.e., oriented surfaces of 

arbitrary genus g ≥ 0. Our main contribution is to show 

that it is possible to extend the local properties of Schnyder 

labeling in a coherent manner to triangulated surfaces. 

We investigate an important class of graphs, namely genus g 

triangulations, corresponding to the combinatorial structure 

underlying manifold triangle meshes of genus g.

Luca Castelli Aleardi, Eric Fusy and Thomas Lewiner 2 

(a) Related Work 

Vertex spanning tree decompositions 

In the area of tree decompositions of graphs there exist 

some works dealing with the higher genus case. We mention 

one recent attempt to generalize Schnyder woods to the 

case of toroidal graphs [5] (genus 1 surfaces), based on a 

special planarizing procedure. In the genus 1 case it is actually 

possible to choose two adjacent non-contractible cycles, 

defining a so-called tambourine, whose removal makes the 

graph planar; the graph obtained can thus be endowed with 

a Schnyder wood. In the triangular case this approach yields 

a process for computing a partition of the edges into three 

edge-disjoint spanning trees plus at most 3 edges. Unfortunately, 

as pointed out by the authors, the local properties of 

Schnyder woods are possibly not satisfied for a large number 

of vertices, because the size of the tambourine might be arbitrary 

large. Moreover, it is not clear how to generalize the 

method to genus g ≥ 2. 

Planarizing surface graphs 

A natural solution to deal with Schnyder woods (designed 

for planar graphs) in higher genus would consist in performing 

a planarization of the surface. Actually, given a surface 

S of genus g and size n, one can compute a cut-graph or 

a collection of 2g non-trivial cycles, whose removal makes 

S a topological disk (possibly with boundaries). There is 

a number of recent works [8, 37, 14, 22, 23, 36] dealing 

with interesting algorithmic challenges concerning the efficient 

computing of cut-graphs, optimal (canonical) polygonal 

schemas and shortest non-trivial cycles. For example 

some works make it possible to compute polygonal schemas 

in optimal O(gn) time for a triangulated orientable manifold 

[23, 36]. Nevertheless we point out that a strategy based 

on such planarizing approach would not be best suited for 

our purposes. From the combinatorial point of view this 

would imply to deal with boundaries of arbitrary size (arising 

from the planarizing procedure), as non-trivial cycles can 

be of size Ω( √ n), and cut-graphs have size O(gn). Moreover, 

from the algorithmic complexity point of view, the 

most efficient procedures for computing small non-trivial cycles 

[8, 22] require more than linear time, the best known 

bound being currently of O(n log n) time. 

Schnyder trees and graph encoding 

One of our main motivations for generalizing Schnyder 

woods to higher genus is the great number of existing 

works in the domain of graph encoding and mesh compression 

that take advantage of spanning tree decompositions 

[21, 30, 34], and in particular of the ones underlying 

Schnyder woods (and related extensions) for planar 

graphs [2, 11, 12, 17, 19, 29]. The combinatorial properties 

of Schnyder woods and the related characterizations (canonical 

orderings [20]) for planar graphs yield efficient procedures 

for encoding tree structures based on multiple parenthesis 

words. In this context a number of works have been 

proposed for the simple compression [19] or the succinct encoding 

[12, 11] of several classes of planar graphs. More recently, 

this approach based on spanning tree decompositions 

has been further extended to design a new succinct encoding 

of labeled planar graphs [2]. Once again, the main ingredient 

is the definition of three traversal orders on the vertices 

of a triangulation, directly based on the properties of Schnyder 

woods. Finally we point out that the existence of minimal 

orientations (orientations without counter-clockwise directed 

cycles) recently made it possible to design the first 

optimal (linear time) encodings for some popular classes of 

planar graphs. For the case of triangulations and 3-connected 

plane graphs [17, 29] there exist some bijective correspondences 

between such graphs and some special classes of 

plane trees, which give nice combinatorial interpretations of 

enumerative formulas originally found by Tutte [35]. Very 

few works have been proposed for dealing with higher genus 

embedded graphs (corresponding to manifold meshes): this 

is due to the strong combinatorial properties involved in the 

planar case. Nevertheless, the topological approach used by 

Edgebreaker (using at most 3.67 bits per vertex in the planar 

case) has been successfully adapted to deal with triangulated 

surfaces having arbitrary topology: orientable manifold with 

handles [26] and also multiple boundaries [25]. Using a different 

approach, based on a partitioning scheme and a multilevel 

hierarchical representation [9], it is also possible to obtain 

an encoding of a triangulation of fixed genus g, having 

f faces and n vertices, requiring 2.175f +O(g log f)+o(f) 

bits (or 4.35n + o(gn) bits) which is asymptotically optimal 

for surfaces with a boundary: nevertheless, the amount of additional 

bits hidden in the sub-linear o(n) term can be quite 

large, of order Θ( n 

log n 

log log n). 

(b) Contributions 

Our first result consists in defining new traversal orders 

of the vertices of a triangulation of genus g, as an extension 

of the canonical orderings defined for planar graphs. 

We are also able to provide a generalization of the Schnyder 

labeling to the case of higher genus surfaces. The major 

novelty is to show that the linear time algorithm designed for 

the planar case can be extended in a nontrivial way in order 

to design a traversal of a genus g surface. This induces 

a special edge coloring and orientation that is a natural generalization 

of the corresponding planar structure. In particular, 

the spanning property characterizing Schnyder wood is 

again verified almost everywhere in the genus g case. Our 

approach involves the implicit computation of cut-graphs, as 

one would expect. Nevertheless, our strategy is really different 

from the one based on planarizing procedures: even if our 

cut-graphs and non-trivial cycles can have arbitrary size, we 

show that it is always possible to propagate the spanning condition 

of Schnyder woods even along this cuts in a coherent 

and natural manner. This property, and in particular the fact 

that there are only few exceptions to the spanning condition, 

allow us to characterize our graph decomposition in terms of 

Preprint MAT. 18/08, communicated on September 1 st , 2008 to the Department of Mathematics, Pontifícia Universidade Católica — Rio de Janeiro, Brazil.

3 Schnyder woods for higher genus triangulated surfaces 

one-face maps of genus g (a natural generalization of plane 

trees). As application, we propose an encoding scheme allowing 

to represent the combinatorial information of a planar 

triangulation of size n with at most 4n + O(g log n) bits. 

This is a natural extension of similar results obtained in the 

planar case [19, 1]. Compared to previous existing works on 

graph encoding in higher genus, our result matches the same 

information theory bounds of the Edgebreaker scheme [30], 

which uses at most 3.67n bits in the planar case, but requires 

up to 4n + O(g log n) bits for meshes with not spherical 

topology [26, 25]. 

2 Preliminaries 

– root face condition: the edges incident to the vertices 

v 0 , v 1 , v n−1 are all ingoing and are respectively of 

color 0, 1, and 2. 

– local condition: For each vertex v not incident to the 

root face, the edges incident to v in ccw order are: one 

outgoing edge colored 0, zero or more incoming edges 

colored 2, one outgoing edge colored 1, zero or more 

incoming edges colored 0, one outgoing edge colored 

2, and zero or more incoming edges colored 1, which 

we write concisely as 

(Seq(In1), Out0, Seq(In2), Out1, Seq(In0), Out2). 

(a) Graphs on surfaces. 

In this work we consider discrete surfaces such as triangular 

meshes, and we consider only the topological information 

specific to such a surface, i.e., the incidences vertices/edges/faces. 

All discrete surfaces we consider can be 

precisely formalized with the concept of map. A map is a 

graph embedded on a surface S such that the areas delimited 

by the graph, called the faces of the map, are topological 

disks. In the following the maps considered have neither 

loop nor multiple edges, which means that the embedded 

graphs are simple. Of particular interest for us are maps 

of genus g with all faces of degree 3; we call such a map 

a triangulated surface of genus g. These maps represent the 

combinatorial incidences of triangular meshes of a surface of 

genus g. Given a surface S, we refer to its primal graph (or 1- 

skeleton) as the underlying graph consisting of its edges and 

vertices. The dual graph of S is defined as the graph having 

the faces as nodes and whose edges correspond to pairs of 

adjacent faces in S. Given a map M on a surface S, acut 

graph of M is a subgraph of M whose induced embedding 

on S defines a one-face map. If S is cut along such a submap 

G c , the surface obtained is equivalent to a topological disk; 

the boundary of the disk contains each edge of G c twice, and 

the original surface can be obtained by identifying and gluing 

together these pairs of edges, which defines a so-called 

polygonal scheme of the surface S (for more details on the 

topological properties of graphs on surfaces see [27]). 

(b) Schnyder woods for Plane Triangulations 

Let us first recall the definition of Schnyder woods for 

plane triangulations, which we will later generalize to higher 

genus. 

While the definition is given in terms of local conditions, 

the main structural property is more global, namely a partition 

of the edges into 3 (essentially) spanning trees (see 

Figure 1). 

Definition 1 ([33]) Let T be a plane triangulation, and denote 

by v 0 ,v 1 ,v n−1 the vertices of the outer face in cw order; 

we denote by E the set of edges of T except those three 

incident to the outer face. A Schnyder wood of T is an orientation 

and labeling, with label in {0, 1, 2} of the edges in 

E so as to satisfy the following conditions: 

A fundamental property concerning Schnyder woods [33] 

is expressed by the following: 

Fact 1 Each plane triangulation T admits a Schnyder wood. 

Given a Schnyder wood on T , the three oriented graphs T 0 , 

T 1 , T 2 induced by the edges of color 0, 1, 2 are trees that 

span all inner vertices and are naturally rooted at v 0 , v 1 , 

and v n−1 , respectively. 

3 Higher genus triangulations 

(a) Generalized Schnyder Woods 

As done in previous works considering the extension 

of Schnyder woods to other classes of (not triangulated) 

planar graphs [15, 29], one main contribution is to propose 

a new generalized version of Schnyder woods to genus g 

triangulations (see Figure 2 for an example). 

Definition 2 Consider a triangulated surface S of genus g, 

having n vertices and a root face (v 0 ,v 1 ,v n−1 ); let E be the 

set of edges of S except those three incident to the root face. 

A genus-g Schnyder wood on the surface S is a partition of E 

into a set of normal edges and a set E s = {e 1 ,e 2 , . . . , e 2g } 

of 2g special edges considered as fat, i.e., made of two 

parallel edges. In addition, each edge, a normal edge or one 

of the two edges of a special edge, is simply oriented and has 

a label in {0, 1, 2}, so as to satisfy the following conditions: 

– root face condition: All edges incident to v 0 , v 1 , and 

v n−1 are ingoing with color 0, 1, and 2, respectively 

(the special edges are allowed to be incident to these 

vertices). 

– local condition for vertices not incident to special 

edges: for every vertex v ∈ S \ ({v 0 ,v 1 ,v n−1 } not 

incident to any special edge, the edges incident to v in 

ccw order are, as in the planar case, of the form: 

(Seq(In1), Out0, Seq(In2), Out1, Seq(In0), Out2). 

– local condition for vertices incident to special edges: 

A vertex v incident to k ≥ 1 special edges has exactly 

one outgoing edge in color 2. Consider the k +1 

sectors around v delimited by the k special edges and 



(a) 

v 

e 2 

w 

e 1 

(b) 

w 

u 

u 

v 0 

v 1 

E s = {e 1 ,e 2 } e 1 e2 

v 

w 

Figure 2: (a) A face-rooted triangulated surface S of genus 1 endowed with a g-Schnyder wood. (b) The local condition for vertices incident 

to special edges. 

the outgoing edge in color 2. Then in each sector, the 

edges occur as follows in ccw order: 

Seq(In1), Out0, Seq(In2), Out1, Seq(In0). 

– Cut-graph condition: The graph T 2 formed by the 

edges of color 2 is a tree spanning all vertices except 

v 0 and v 1 , and rooted at v n−1 with all edges of the tree 

oriented toward the root. The graph C formed by T 2 

plus the 2g special edges is a unicellular map (i.e., a 2- 

cell embedded graph with a unique face); C is called 

the cut-graph of the Schnyder wood. 

The edge-based characterization of a triangulated surface 

proposed above has some advantages that make it a natural 

generalization of the planar definition. As one would expect, 

the planar and the genus g definitions do coincide when considering 

planar triangulations. In the planar case (g =0), the 

last condition states that T 2 is a tree, which is actually already 

implied by the local conditions (which is not the case 

in higher genus). Second we observe that the local conditions 

hold at every vertex except the extremities of the 2g 

edges in E s , where very similar conditions are satisfied. In 

particular almost all the vertices have out-degree 3, which 

makes the g-Schnyder woods a good characterization of the 

local planarity of a bounded genus surface. Finally, we point 

out that the local condition above leads to graph decompositions 

expressed in terms of one-face maps having genus g, 

which are the natural generalization of plane spanning trees 

(see Proposition 13 and the remark after). 

(b) Handle operators 

Following the approach suggested in [26], based on Handlebody 

theory for surfaces, we design a new traversal strategy 

for higher genus surfaces: as in the planar case, our strategy 

consists in conquering the whole graph incrementally, 

face by face, using a vertex-based operator (conquer) and 

two new operators (split and merge) designed to represent 

the handle attachments. We start by setting some notations 

and definitions. We consider a triangulated surface 

S having genus g and n vertices. We denote by S in (S out ) 

the embedded growing subgraph of S induced by the faces 

already conquered (not yet conquered, respectively). S out 

is a face-connected map of genus g having b ≥ 1 boundaries, 

each boundary being a simple cycle C i , i ∈ [1..b]. 

We define ∂S in := ∪ b i=1 C i as the overall border between 

S in and S out . By design of our algorithm, the root-face 

{v 0 ,v 1 ,v n−1 } is always in S in and the edge (v 0 ,v 1 ) is always 

on ∂S in ; this edge is called the base-edge of the boundary. 

Handle operator of first type 

Definition 3 A chordal edge is an edge of S out \∂S in whose 

two extremities are on ∂S in . A boundary vertex w ∈ C i is 

free if w is not incident to a chordal edge e. 

We can now introduce the first operator, called conquer, 

which adds a free vertex w with all its incident edges and 

faces to S in . More precisely, the conquest or removal of a 

vertex w consists in attaching a set of triangles to S in as 

follows (see the first picture in Figure 3): 

• conquer(w), with w a free vertex: update the conquered 

area, by transferring from S out to S in all faces incident 

to w that were not yet in S in . The vertices and edges of 

S out (S in ) are naturally updated as those vertices and edges 

incident to the faces of S out (S in , respectively). 

This operation corresponds to a handle operator of type 1 

(see [26]), not modifying the topology of S in . 

Handle operators of second type 

A chordal edge e for S out is said to be non-separating 

if S out is not disconnected when cutting along e. A chordal 

edge whose two extremities are in the same cycle C i is called 

a splitting chordal edge. For such an edge, we define C ′ and 

C ′′ as the two cycles formed by C i + e. Then e is said to 



w l 

w 

w r 

C 

S out S in v 0 

v 1 

v 2 

S in 

w 

w 

u 

S out 

v 2 

S out 

S in u v 0 

v 1 

S out v 0 v 1 

v k 

v k 

v k 

conquer(w) chordal edge (u, w) 

split(u, w) 

S in 

v 2 

S in 

S in 

S in 

w 

S out 

v 0 

v 1 

v 2 

merge(u, w) 

S out 

u 

S in 

Figure 3: Illustrated on a toroidal graph, from left to right, a conquer operation, a contractible chordal edge, a split edge, and a merge 

edge. 

be contractible if either C ′ or C ′′ is contractible. Note that a 

non-separating splitting chordal edge can not be contractible. 

Definition 4 (split edge) A split edge for the area S out is a 

non-separating splitting chordal edge. 

It is easy to see that the addition of a split edge to S in 

(together with a thin band B enclosing the edge) does not 

change the genus of S in and increases by one the number of 

components (cycles) on the boundary of S in , i.e., the Euler 

characteristic of S in decreases by 1. The addition of B to 

S in (and removal of B from S out ) is called a split. 

Definition 5 (merge edge) A merge edge for the area S out 

is a non-separating chordal edge e having its two extremities 

on two distinct cycles C i and C j , i ≠ j. 

This time the addition of a merge edge to S in (together 

with a thin band B enclosing the edge) adds a handle to 

S in , hence increases the genus of S in by 1. In addition, 

two components of the boundary of S in are merged, hence 

the number of components (cycles) of the boundary of S in 

decreases by 1. Therefore the Euler characteristic of S in is 

now χ ′ =2− 2(g + 1) − (b − 1) = 2 − 2g − 1, i.e., has 

decreased by 1. The addition of B to S in (and removal of B 

from S out ) is called a merge. 

The edges involved in the merge/split operations are obviously 

the merge edges and split edges. These will be exactly 

the special edges of the genus g Schnyder wood to be 

computed by our traversal algorithm. The vertices incident 

to merge or split edges are called multiple vertices, as each 

merge/split operation results in considering such a vertex as 

duplicated. 

(c) Computing a Schnyder wood of a planar triangulation 

In this section we briefly review a well-known linear time 

algorithm designed for computing a Schnyder wood of a planar 

triangulation, following the presentation by Brehm [7]. 

Once defined the conquer operation, we can associate to it 

a simple rule for coloring and orienting the edges incident to 

a vertex conquered: 

• colorient(v): orient outward of v the two edges connecting 

v to its two neighbors on ∂S in ; assign color 0 (1) to 

the edge connected to the right (left, respectively) neighbor, 

looking toward S out . Orient toward v and color 2 all edges 

incident to v in S out \∂S in . 

We can now formulate the algorithm for computing a 

Schnyder wood as a sequence of n − 2 conquer and 

colorient operations, following a simple and elegant presentation 

adopted by Brehm [7]. Given a plane triangulation 

T with outer face {v 0 ,v 1 ,v n−1 } we start with S in —the set 

of visited vertices— initialized to {v 0 ,v 1 ,v n−1 }, and with 

∂S in —the border of S in — initialized to be the contour of 

the outer face. 

COMPUTESCHNYDERPLANAR(T ) (T a plane triangulation) 

while C ≠ {v 0 ,v 1 } 

Choose a free vertex v on C; 

conquer(v); colorient(v); 

end while 

Correctness 

The correctness and termination of the traversal algorithm 

above is based on the fundamental property that an edgerooted 

triangulated topological disk always has on its boundary 

a vertex not incident to any internal chord and not incident 

to the root edge, e.g., a free vertex different from v 0 and 

v 1 (see [7] for a detailed proof). 

One major advantage of computing a Schnyder wood in 

an incremental way is that we are able to put in evidence 

some invariants, which remain satisfied at each step of the 

algorithm: 

– the edges that are already colored and oriented are 

those in S in \∂S in ; 

– for each vertex v ∈ S in \∂S in , all edges incident 

to v are colored and directed in such a way that the 

Schnyder rule is always satisfied; 

– every boundary node v ∈ C \{v 0 ,v 1 } has exactly 

one outgoing edge lying in S in \∂S in . This edge e has 

color 2, and the ingoing edges in S in \∂S in incident to 



v n−1 

v 0 v 1 

Figure 4: Some steps of the algorithm that computes a Schnyder wood of a planar triangulation T (rooted at (v 0 ,v 1 )). The traversal is 

performed as a sequence of conquer operations on free vertices: edges in S out (dark region) are oriented and directed according to the 

Schnyder rule. 

v between e and the right (left) neighbour of v, looking 

toward S out , have color 1 (0, resp.). 

Using these invariants, it is straightforward to prove the 

following lemma (see [7] for a detailed presentation): 

Lemma 6 Given a planar triangulation T with outer face 

(v 0 ,v 1 ,v n−1 ) the algorithm COMPUTESCHNYDERPLANAR 

computes in linear time a Schnyder wood of T . 

(d) A new traversal for genus g surfaces 

This section presents our main contribution, an algorithm 

for traversing a triangulated surface S of arbitrary genus 

g ≥ 0, which extends the algorithm of Brehm in a natural 

way. As in the planar case, the traversal is a greedy sequence 

of conquer operations on free vertices, with here 

the important difference that these operations are interleaved 

with 2g merge/split operations so as to make the genus 

of the conquered area increase from 0 to g. We start with 

C := {C 0 } := {(v 0 ,v 1 ,v n−1 )}. At the beginning S in is 

a topological disk delimited by the simple cycle C 0 . As in 

the planar case, we make use of the operation conquer(v) 

that consists in transferring the (remaining) faces incident to 

v from S out to S in . Associated with such a conquest is the 

colorient rule for coloring and orienting the edges that 

are conquered, as defined in Section 3(c). We also make use 

of the handle operations split and merge, as defined in 

Section 3(b). 

COMPUTESCHNYDERANYGENUS(S) (S a triangulated 

surface of genus g) 

while {C} ≠ {v 0 ,v 1 } 

If there exists a free vertex v on 

some C i ∈C 

conquer(v); colorient(v); 

otherwise, if there exists a split 

edge e =(u, w) ∈ S out \∂S in ; 

split(u,w); thereby adding a new 

cycle C ′ to C; 

otherwise, find a merge edge e =(u, w) ∈ 

S out ; 

merge(u,w); thereby removing a cycle 

from C; 

end while 

Figure 7 shows the traversal algorithm executed on a 

toroidal triangulation. 

Notice the subtlety that, for nonzero genus, the multiple 

vertices, i.e., the vertices incident to merge/split edges, 

are conquered several times, as illustrated in Figure 6. Precisely, 

for a vertex v incident to k ≥ 0 merge/split edges, 

conquer(v) occurs k +1times. 

Our main result is expressed by the following 

Theorem 7 Any triangulated surface S of genus g admits 

a genus g Schnyder wood, which can be computed in linear 

time. 

The validity of this property relies on the Lemmas stated 

in the next section. 

(e) Termination and complexity of the algorithm 

Lemma 8 For any triangulated surface S of genus g, 

COMPUTESCHNYDERANYGENUS(S) terminates. 

Proof : 

It suffices to show that a merge or split operation is always 

possible whenever there remain no free vertices on 

∂S in . The main idea is to use an induction argument (on the 

number of faces of S out , which is decreasing along the conquest) 

and to proceed by contradiction to show the existence 

of non-contractible non-separating cycles corresponding to 

split/merge edges. More precisely, assume by contradiction 

that we have an example of situation where we are stuck, i.e., 

no free vertex on the boundary, and only contractible chordal 

edges or non-contractible separating chordal edges. Assume 

also that this is the smallest example of such a situation in 

terms of the number of faces in the non discovered area (notice 

that this number is at least 2 otherwise we would be at 

the last step of the algorithm where there is no problem; one 

just conquer the vertex not incident to the root edge in the 

last remaining not conquered face). Let us first observe that 

there cannot be a contractible chordal edge, since it would 

yield a free vertex on ∂S in , precisely, a free vertex in the 

sub-path P of ∂S in such that P + e is a contractible cycle 

(this is due to the well known property that a triangulated dissection 

of a polygon, with a base edge on the polygon, has at 

least one vertex not incident to the base edge nor incident to 

any internal chord, see [7]). Hence, all edges in S out \∂S in 

have to be non-contractible separating edges. Let e be such 



(planar case) 

w 

C 

C 

S out 

invariants 

S out 

conquer(w)+ 

colororient(w) 

Figure 5: These pictures show the result of a colorient operation in the planar case. 

an edge. Cutting S out along e consists in splitting into two 

smaller surfaces S 1 and S 2 , each of which is partitioned into 

a conquered area and a not conquered area. Assume w.l.o.g. 

that S 1 is the area whose boundary (between conquered and 

not conquered) does not contain the base-edge; and now define 

e as the base-edge of that boundary. Any free vertex v 

for S 1 is also a free vertex for S (indeed by definition v is 

not incident to any chord for S 1 nor is it an extremity of e; 

hence v is not incident to any chord of S). Therefore, as S 

has no free vertex, S 1 has neither a free vertex. Now observe 

that a non-contractible non-separating cycle for S 1 is also 

non-contractible and non-separating for S. Hence there can 

not be any candidate for a merge or split in S 1 (since there is 

no such candidate in S). We have thus reached a contradiction, 

as S 1 has fewer faces than S in its not conquered area 

(because of the minimality of S). 

 

Lemma 9 For any triangulated surface S of 

genus g having n vertices, the procedure 

COMPUTESCHNYDERANYGENUS(S) can be implemented 

to run in O(n) time. 

Proof : Implementing the algorithm to run in linear time requires 

some bookkeeping. For this purpose, we maintain the 

list of chordal edges for ∂S in , and, which is more difficult, 

we maintain a classification of these chordal edges as contractible, 

non-contractible separating, split edges, and merge 

edge. Such a classification is done by maintaining a spanning 

cut-graph G c for the dual of S out , such that G c contains the 

dual of all chordal edges; and by maintaining a depth-firstsearch 

spanning tree of G c so as to test which edges of G c 

disconnect G c . All this bookkeeping can be done in amortized 

linear cost, as detailed below. Firstly, consider a special 

vertex spanning tree T p of S out , including the boundary 

∂S in in the following way (blue edges in Fig. 8). For 

each boundary component C i of ∂S in choose a boundary 

edge (x i 0,x i 1) (where for the first component we take the 

root edge (v 0 ,v 1 )) and add all the edges in ∂S in \ C i to the 

set T p (which initially consists of a set of disjoints paths). 

Then perform a depth-first traversal S out starting from v 0 , 

that computes a spanning tree of the primal graph of S out : 

add its edges to T p . Note that by construction chordal edges 

do not appear in T p , as each extremity already lies to the 

paths added in the first step. Now consider the dual map G c 

(red edges in Fig. 8) of T p , which is a sub-map of the dual 

graph of S out , defined in the following manner: its nodes 

correspond to the faces of S out , while its arcs are exactly 

the dual of that edges not appearing in T p . If we denote by 

g ′ the genus of S out , then G c is a map of genus g ′ , having 

only one face. We firstly simplify this structure, by recursively 

eliminating all degree one nodes (vertices which are 

leaves at some step of this decimating procedure). Note that 

this transformation does not affect the topology of the resulting 

map obtained by G c , which has still genus g ′ and 

one face, thus containing 2g ′ non trivial cycles. Observe that 

each chordal edge has a corresponding dual arc in G c . We 

now compute a vertex spanning tree of such map, which will 

contain ‖G c ‖−2g ′ arcs (as it cannot contain any cycle, by 

construction). Note that the cycle contained in map G c must 

be non-trivial (non-contractible and non-separating). For example, 

the existence of a contractible cycle would imply the 

existence of a vertex v which would not appear in T p (which 

by construction is a spanning tree of all the vertices of S out ). 

On the other hand, the existence of a non-contractible but 



(toroidal case) 

w 

S in 

w 

S in 

w 

S in 

w 

S in 

S out 

u 

S out 

S out 

u 

S out 

S out 

u 

S out 

S out 

u 

S out 

S in 

S in 

S in 

S in 

split(u, w) 

conquer(w)+ 

colorient(w) 

conquer(u)+ 

colorient(u) 

conquer(u)+ 

colorient(u) 

Figure 6: These pictures show the result of colorient operations in the higher genus case. Any chordal split (or merge) edge (u, w) can 

be oriented in one or two directions (having possibly two colors), depending on the traversal order on its extremities. 

separating cycle would force the region S in to be disconnected. 

So, the 2g ′ missing arcs belonging to G c allows to get 

classification of the arcs of G c : in particular we can distinguish 

separating arcs (which disconnect G c ), from those arcs 

belonging to non-trivial cycles. This translates into a classification 

of the chordal edges in S out : the special edges we look 

for are exactly the chordal edges dual of the arcs belonging to 

non-trivial cycles. The steps above provide thus a linear time 

procedure for detecting split and merge edges. Such a procedure 

must be run at most O(g) times, since we need to locate 

a special edge only after all free boundary vertices have been 

treated, which guarantees the overall linear time complexity 

of our algorithm for computing a g-Schnyder wood. 

(f) The algorithm computes a genus g Schnyder wood 

Lemma 10 Let S be a triangulated surface of genus g. 

Then the edge orientation and coloration computed by the 

procedure COMPUTESCHNYDERANYGENUS(S) satisfy the 

local conditions of a g-Schnyder wood for S. 

Proof : To prove that the combinatorial structure obtained 

by the traversal algorithm is a genus g Schnyder wood, it 

suffices to define some invariants that remain satisfied all 

along the traversal. We make use of the invariants already 

defined for the planar case (Section 3.c(i)), as well as new 

invariants relative to multiple vertices, which state that the 

local condition is satisfied for multiple vertices in S in \∂S in , 

and is already partially satisfied for those on ∂S in . More 

precisely, let w be a multiple vertex incident to a special 

edge e 1 =(u, w), and consider a sector of edges incident 

to w, defined by the pair of special edges (e 1 ,e 2 ), where 

e 2 =(w, z). Then, assuming the outgoing edge of color 2 

incident to w belongs to the sector above, we have: 

– the incident edges belonging to the sector defined by 

e 1 and e 2 are listed around w follows (starting from 

e 1 , in cw order). Zero or more ingoing edges of color 

1 (possibly including e 1 ), one or more non-oriented 

and non-colored edges, zero or more ingoing edges of 

color 0, one outgoing edge of color 2, zero or more 

ingoing edges of color 1, one ore more non-oriented 

and non-colored edges, zero or more ingoing edges of 

color 0. 

This completely describes the orientation and coloration 

of edges around w before its conquest: recall that a multiple 

vertex incident to k special edges is conquered more than 

once, and it does disappear from the boundary after exactly 

k +1conquer(w) operations. It now remains to show 

that the conquest of neighboring vertices does not affect the 

invariant above: we will consider several cases, depending 

on the type of boundary vertex (multiple or not) on which 

we perform the conquest. First, let w r be the right neighbor 

of w on ∂S in : performing conquer(w r ) consists in orienting 

the edge (w, w r ) toward w, with color 0. The number of the 

outgoing edges remains unchanged, and the invariant above 

is still valid. The remaining case, concerns the conquest of 

the left neighbor of w, say w l (possibly coinciding with u). 

Performing conquer(w l ) consists in adding an edge of color 

1, oriented toward w. In this case we only increases of 1 

the number of consecutive ingoing edges of color 1, which 

are first encountered when turning around w starting from 

(u, w) in cw order. Again, the invariant above is satisfied. To 

conclude our proof, is straightforward to see that performing 

a conquer(w) operation does not affect the Schnyder local 

rule around w. Just observe that: the leftmost incident edge 

will be oriented outgoing with color 0 (and is preceded 

by zero or more ingoing edges of color 1, because of the 

invariant above); the rightmost edge will be outgoing of color 

1 (being followed in cw order by a sequence of zero or more 

ingoing edges of color 0); and the remaining incident edges 

are oriented toward w having color 2. Note that after each 

conquer(w) operation the number of outgoing edges of 

color i (i =0, 1) increases by 1, according to definition 2. 

In a similar way, we deal with the conquest of w and of its 



S 

S out S in 

e m 

w ′ e m 

v w 

u ′ 

u 

v w 

w ′′ 

e s 

u 

w 

e s 

e m 

Figure 7: An example of execution of our traversal algorithm for a triangulated surface of genus 1. As in the planar case the traversal starts 

from the root face (the one incident to the green root edge). As far as only conquer operations on free vertices are performed, the area already 

explored (white triangles) remains planar and homeomorphic to a disk (second picture). When there remain no free vertices to conquer (third 

picture), it is possible to find split and merge edges (incident to black circles). After performing split and merge operations, (here on the edges 

(u, w) and (w, v)), new free vertices can be found (here u ′ , w ′ , and w ′′ ) in order to continue the traversal, until the entire graph is visited. 

neighboring vertices lying in the opposite side of the sector 

(e 1 ,e 2 ), respect to the edge e 2 (recall that e 1 and e 2 could 

coincide, when w has multiplicity 2 being incident to only 

one special edge). 

 

Consider a g Schnyder wood computed by our algorithm, 

and let C be the cut-graph of the Schnyder wood. In the 

statement of the following lemma, let C be the part of 

C already conquered at some step of the algorithm (the 

sub-graph contained in S in ). Each connected area of S\C 

is called a face of C (though C might not be a cellular 

embedding, i.e., some of these faces might not be topological 

disks). 

Lemma 11 At each step (strictly before the end) of the algorithm 

computing a g Schnyder wood, let C be the part of 

the cut graph already discovered. Then there is exactly one 

cycle of the boundary of S in inside each facial walk of C. 

Moreover S in \C is a planar surface. 

Proof : The property is true initially: S in is the root face, 

which is planar, and C is reduced to v n−1 , which has the 

total surface S as its unique face. The property remains also 

true at each vertex conquest, since each conquest simply 

increases the area of S in within a facial walk W of C, but 

does not close the boundary of S in within W unless we are 

at the end of the algorithm. If the boundary is closed, the 

area of the surface S within W will be a face in C, hence 

that face must be the unique face of C, i.e., the algorithm 

is at its end. Otherwise the interior of that face would be a 

contractible area, so the final cut-graph would have a face 

with a contractible boundary, which is impossible. For each 

split, we cut a cycle C of the boundary of S in into two 

parts C 1 and C 2 , but the special edge addition also cuts the 

corresponding facial walk W of C into two facial walks 

W 1 and W 2 that contain respectively C 1 and C 2 . The genus 

of S in does not change during a split, so S in \ C remains 

planar. For each merge, we merge two cycles C 1 and C 2 of 

the boundary of S in into a unique cycle by means of a merge 

edge that establishes a bridge between the two components. 

The corresponding facial walks W 1 and W 2 of C are also 

merged into a unique facial walk W by means of the special 

edge, and W contains C in its interior, as illustrated in 

Figure 10. It is also easily observed on that figure that the 

two boundary cycles previously enclosed inside W 1 and W 2 

have been merged into a unique boundary cycle inside W . 

Hence the surface S in \ C remains planar. 

 

Proposition 12 The algorithm COMPUTESCHNYDERANYGENUS(S) 

computes a g-Schnyder wood. 

Proof : As previously seen, the algorithm terminates and 

the orientations and colorations of edges satisfy the local 

properties of a g-Schnyder wood. It remains to show the cutgraph 

property. First let us show that T 2 is a tree spanning all 

vertices except v 0 and v 1 . By the local conditions, all vertices 

except those of the root face have one outgoing edge of color 



S in 

S in 

S in S in 

spanning tree T p dual map G c G c after simplification 

S in 

Figure 8: Special edges (dual of green edges) can efficiently be found by computing a spanning tree of the map G c . 

2, and the only sink for ingoing edges of color 2 is v n−1 . 

Hence to prove that T 2 is a tree rooted at v n−1 it suffices to 

prove that T 2 is connected, precisely that each vertex is the 

origin of a path in T 2 going to the root v n−1 . This property 

is clearly maintained for the vertices of the conquered area 

all along the algorithm. Indeed each time a vertex w appears 

in S in , it is due to the conquest of a vertex v previously on 

the boundary of S in . In addition by definition of a conquest, 

there is an edge of color 2 from w to v. Hence the path of 

color 2 from v to v n−1 is extended to a path from w to v n−1 . 

Let us now prove that the graph C formed by T 2 and the 

special edges is a unicellular map on the surface. Just before 

the last step of the algorithm (at that time the entire cutgraph 

is discovered), the unique face t not in S in is the face 

incident to the base edge on the other side of the root face. 

Hence ∂S in has a unique component (cycle) and this cycle is 

contractible. According to Lemma 11, S in \C is planar and 

C has a unique face. The interior of C is the union of a planar 

annulus (S in ) and of the triangular face t. Hence the interior 

of C is a topological disk, i.e., C is a unicellular map. 

4 The maps in color 1 and 2 are also cellular 

In this section we show that g-Schnyder woods provide a 

characterization in terms of a special class of genus g spanning 

maps, which can be viewed as a good generalization of 

plane trees. 

Proposition 13 Consider a genus g Schnyder wood computed 

by the algorithm COMPUTESCHNYDERANYGENUS. 

Then the embedded graph T 0 (T 1 ) formed by the edges in 

color 0 (1, resp.) is cellular and has at most 1+2g faces. 

Proof : When doing all the steps of COMPUTESCHNYDER- 

ANYGENUS, in reverse chronological order, one grows the 

surface S out until it occupies all the surface S except the root 

face. During that process, let T 1 be the (growing) part of T 1 

already discovered (i.e., the part of T 1 in S out ). To prove the 

result, it suffices show the following invariant when growing 

S out : S out \T 1 remains planar, each boundary of S out is enclosed 

inside a unique facial walk of T 1 , and each facial walk 

of T 1 contains at most one boundary of ∂S in . Let us first describe 

the effect of conquest, split, and merge on S out when 

doing these operations in reverse chronological order. One 

checks that a (reverse) conquest adds a vertex on the boundary 

of S out , and does not change the topology (number of 

boundary components and genus) of S out . In addition it is 

well known in the study of topological surfaces [31] that a 

(reverse) split is also a split for S out and that a reverse merge 

is also a merge for S out . The only difference is that the two 

bases of the bridge-band for S in are the sides of the bridgeband 

for S out and the two sides of the bridge-band for S in 

are the bases of the bridge-band for S out . In each case, the 

conquests performed at the vertices on each side of the thin 

bridge-band create a bridge of edges of color 1 traversing the 

band transversally. By an argument similar to the one used in 

Lemma 11 (see also Figure 10) this ensures that S out \T 1 remains 

planar and, more generally, that the invariants stated 

above still hold. 

Finally, the bound 1 + 2g on the number of faces is due to 

the fact that each special edge contributes at most by one to 

the number of new faces of T i , for i =0, 1 (see Fig. 12). 

Remark 1 The properties of T 0 ,T 1 ,T 2 stated in Proposition 

13 can be considered as extensions of the fundamental 

property of planar Schnyder woods, which states that T 0 , 

T 1 , and T 2 are plane trees. Figure 11 shows an example in 

genus 1. 

5 Application to encoding 

In the planar case, Schnyder woods yield a simple encoding 

procedure for triangulations, as described in [19] and 

more recently in [1]. Precisely, a Schnyder wood with n 

inner vertices is encoded by two parenthesis words W, W ′ 

each of size 2n, where W is the classical Dyck encoding 

for the tree T 2 ; and W ′ is obtained from a clockwise walk 

around T 2 starting at v n−1 , writing an opening (closing) 

parenthesis each time an outgoing edge in color 1 (ingoing 

edge in color 0, resp.) is crossed, and completing the end 

of the word with closing parenthesis. Hence, for a triangulation 

with n +3vertices, the coding word has length 4n. 

This code is both simple and quite compact, as the length 

4n is not far from the information-theory lower bound of 

( 

log2 

( 

4 4 /3 3) · n ) ≈ 3.245 · n, which is attained in the planar 

case by a bijective construction due to Poulalhon and 

Schaeffer [29]. 

Such a bijective construction is still to be found for higher 

genus, but we can here at least extend to higher genus the 

simple encoding procedure based on Schnyder woods. To 



S in 

S out 

w r 

w 

w 

z 

S in S in 

u 

z 

u 

conquer(w r )+ 

colorient(w r ) 

w l 

w 

z 

u 

conquer(u)+ 

colorient(u) 

w 

z 

u 

conquer(w l )+ 

colorient(w l ) 

w 

z 

u 

conquer(w)+ 

colorient(w) 

w r 

conquer(w l )+ 

Figure 9: The correctness of our construction relies on a new invariant describing the edge orientation around multiple vertices. 

encode the Schnyder wood we proceed in a similar way 

as in the planar case except that we have to deal with the 

special edges. First, the one-face map C formed by T 2 and 

the special edges (the cut-graph of the Schnyder wood) is 

encoded by the Dyck word W for T 2 , augmented by O(g) 

pointers, each of size O(log(n)) bits, so as to locate the 

two extremities of each special edge. We obtain the second 

encoding word W ′ from a clockwise walk along the unique 

face of C starting at v n−1 , writing an opening (closing) 

parenthesis each time an outgoing edge in color 1 (ingoing 

edge in color 0, resp.) is crossed, and completing the end of 

the word with closing parenthesis. 

Lemma 14 Let S be a triangulated surface endowed with 

a g-Schnyder wood. Then the information given by edges 

of color 0 is redundant, i.e., the Schnyder wood can be 

recovered even after all edges of color 0 have been deleted. 

Proof : When all edges of color 0 are deleted, we obtain a 

map M on the surface with only edges of color 1 and 2 (and 

the special edges). In each face f of this map, there is one 

corner that received all ends of the edges of color 0 (before 

they were deleted) that were inside f. This corner can be 

recovered since it is the unique corner around the face such 

that the right edge of the angle (looking toward the interior 

of f) is outgoing of color 1 and the left edge of the angle is 

outgoing of color 2. 

 

Lemma 15 Consider a genus g Schnyder wood computed 

by the algorithm COMPUTESCHNYDERANYGENUS, whose 

cut-graph is unfolded into a flat polygon. Let e = (u, v) 

be an edge of color 1 (oriented from u to v). Then u is 

encountered before v during a cw walk along the polygon 

(i.e., with the interior of the polygon on its right) starting at 

v n−1 . 

Proof : Among the two walks along the polygon that join 

u and v, consider the one, called W e , that does not pass by 

v n−1 . Proving the lemma is equivalent to proving that the 

region enclosed by e and W e is on the left of e. Now consider 

the step of the algorithm where u is conquered. Before that 

step, e is on a certain component (cycle) C of the boundary 

∂S in . By definition of a conquest, e has the conquered area 

on its left when it is treated. According to Lemma 11, C is 

enclosed inside a face f of the current cut-graph C. Let W e 

be the walk around f from u to v. Clearly the area enclosed 

by e and W e is on the left of e, and W e does not pass by 

v n−1 (to be proved). In addition, this walk along C is now 

completely comprised inside the conquered region after the 

conquest involving e, hence it will not change until the end of 

the algorithm, i.e., W e will still be the unique walk along the 

final cut-graph C joining u and v without passing by v n−1 . 

This concludes the proof. 

 

Lemma 15 is crucial to encode the edges of color 1 (red 

edges). When walking around the cut-graph (with the face 

on the right), write an opening parenthesis when crossing 

an outgoing edge of color 1 and a closing parenthesis when 

crossing an ingoing edge of color 1. According to the lemma 

above, the word we obtain is a parenthesis word (each prefix 

has at least as many opening as closing parentheses) and 

each matching corresponds to an edge of color 1. In addition, 

from the parenthesis word, it is easy to place the ingoing 

half-edge and outgoing half-edges of color 1 (before doing 

the matchings so as to form complete edges of color 1). Indeed, 

one can recognize around the cut graph (using the local 

conditions of Schnyder woods) which corners can receive ingoing 

edges of color 1 and which corners can receive ingoing 

edges of color 1. 

To conclude, the edges of color 0 are redundant, the cutgraph 

can be encoded by a parenthesis word of length 2n (for 

the tree T 2 ) plus O(g) pointers each of length O(log(n)) (for 

the special edges), and the edges of color 1 can be encoded 

by a parenthesis word of length 2(n +4g) (since there are 

n +4g edges of color 1, each of which corresponds to a 

matching of the parenthesis word). All in all, we obtain the 

following result: 

Proposition 16 A triangulated surface of genus g with n+3 

vertices can be encoded — via a genus g Schnyder wood — 

by a binary word of length 4n + O(g log(n)). Coding and 

decoding can be done in linear time. 

We mention that one could also design a more sophisticated 

code that supports queries: this time we should need to 

encode the three maps T i using multiple parenthesis systems 

(as done in [12, 2]). 



a) 

b) 

Figure 10: The effect of a merge operation on the conquered surface and on the cut graph. Two faces of the cut graphs are merged and two 

boundaries of the conquered surface are merged. 

Given a (unlabelled) triangulation S of genus g with e 

edges, it is possible to obtain a compact representation of S 

using asymptotically (2 log 2 6) · e + O(g log e) bits which 

answers queries for vertex adjacency and vertex degree in 

O(1) time. Once again the main idea is to compute a g- 

Schnyder wood of S and to encode the corresponding maps 

T i , i ∈{0, 1, 2}. This time, in order to efficiently support 

adjacency queries on vertices we have to encode the three 

maps T i using several types of parentheses: we use a multiple 

parenthesis system (with 2 types of parentheses) as above 

to describe each map T i (the arguments are similar to the 

ones given in [2], where the planar labeled triangulations 

are considered). We mention that the partitioning strategy 

presented in [9] achieves a better bound of 2.175 bits per 

face, more O(g log n) additional bits for dealing with handles 

(with different face based navigation). Nevertheless, our 

approach could be successful in extending existing encodings 

for more general planar graphs ([12]) and for labeled 

graphs ([2]) to the higher genus case. 

6 Conclusion and perspectives 

We have presented a general approach for extending to 

higher genus a fundamental combinatorial structure, Schnyder 

woods, which is by now a standard tool to handle planar 

graphs both structurally and algorithmically. We have been 

successful in showing that the definition and several fundamental 

combinatorial properties can be extended from the 

planar to the genus g case in a natural way. We think that this 

work is a first step in the attempt of generalizing the properties 

of Schnyder woods and more generally α-orientations 

(orientations with prescribed out-degrees) to other interesting 

classes of graphs. 

Canonical orderings and Schnyder Woods 

We point out that our graph traversal procedure induces 

an ordering for treating the vertices so as to shell the surface 

progressively. Such an ordering is already well known in the 

planar case under the name of canonical ordering and has 

numerous applications for graph encoding and graph drawing 

[12, 20]. It is thus of interest to extend this concept to 

higher genus. The only difference is that in the genus g case 

there is a small number —at most 2 · 2g— of vertices that 

might appear several times in the ordering; these correspond 

to the vertices incident to the 2g special edges (split/merge 

edges) obtained during the traversal. There are several open 

questions we think should be investigated concerning the 

combinatorial properties of such orderings and the corresponding 

edges orientations and colorations. For example, 

one may ask whether to any edge coloration and coloring 

defining a Schnyder wood could correspond a canonical ordering, 

as in the planar case. We think also it would be interesting 

to see whether such an ordering would yield an efficient 

algorithm for drawing a graph on genus g surface (as it 

has been done in the planar case [20]). 

Further extensions 

Our approach relies on quite general topological and combinatorial 

arguments, so the natural next step should be to 

apply our methodology to other interesting classes of graphs 

(not strictly triangulated), which have similar characterization 

in the planar case. Our topological traversal could be 

extended to the 3-connected case, precisely to embedded 3- 

connected graphs with face-width larger than 2, which correspond 

to polygonal meshes of genus g. We point out that 

our encoding proposed in Section 5 could take advantage of 

the existing compact encodings of planar graphs [12, 11, 19], 

using similar parenthesis-based approaches. 

Lattice structure and graph encoding applications 

There are a number of deep and interesting questions 

that our work leaves open. From the combinatorial point 

of view it should be of interest to investigate whether edge 

orientations and colorations in genus g have nice lattice 

properties, as in the planar case. In particular, the existence of 

minimal orientations in the planar case led to a large number 

of deep results in areas such as graph drawing, optimal 

sampling and coding. 

In addition to the new tree decomposition proposed 

for the higher genus case, one contribution has been to 

show, from the algorithmic point of view, that a vertexdecomposition 

based traversal can be performed in linear 

time for any genus g ≥ 0. In particular, the linear time complexity 

of our traversal is due to an efficient procedure for 

checking whether a chordal edge is non-contractible. Thus 



S 

v 

v 

e m 

u 

w 

w 

e s 

v w 

v w 

v e m w 

F 1 

F 1 

F 3 

e 

F 1 F F 2 

F 

F s 

1 

1 

1 

u 

F 1 

u 

u 

F 1 F 1 

F 1 

F 1 

F 1 

F 1 

v F 2 w 

v w 

v w 

F 1 

T 1 T 0 T 2 

Figure 11: A triangulated torus endowed with a Schnyder wood. The black edges form a tree T 2 , and the addition of the two special edges 

yields a one-face map. The embedded graph T 0 (blue edges) is a map with 3 faces; and the embedded graph T 1 (red edges) is a map with 2 

faces. 

a “left-most” traversal of the triangulated surface should be 

doable in linear time (let us recall that a left-most driven 

traversal returns in the planar case the minimal orientations 

mentioned above). This remark, together with the local properties 

of genus g Schnyder woods (Definition 2), suggests 

the possibility of further nice applications in graph encoding 

and sampling (see [17, 29] for the planar case). More 

precisely, we conjecture the existence of correspondences 

between some special classes of one-face maps (generalizations 

of plane trees) and the classes of triangulated and polyhedral 

combinatorial meshes of genus g: this would lead to 

the first (linear time) optimal compression schemes and succinct 

encodings for these fundamental classes of meshes. 

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w 

S in 

S in 

S in 

w 

w w a 

F 2 

F 2 

w a 

S out S out S out 

u v 

S out v 0 0 u v 0 

u 

u a 

v 1 

v 1 

S in v 1 

S in 

S out 

S in 

S out 

Figure 12: This pictures illustrate how the double orientation of special edges can contribute to the additional (trivial) faces to maps T 0 

(or T 1 ): in this pictures the double orientation of (u, w) adds a face F 2 to the map T 1 . Depending on its coloration, (u, w) can (or not) be 

included on the boundary of F 2 

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Schnyder woods for higher genus triangulated surfaces with ...

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