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Triangle Contact Representations and Duality?

Daniel Gonc ¸alves, Benjamin L´ evˆ eque, and Alexandre Pinlou

LIRMM, CNRS & Univ. Montpellier 2

161 rue Ada 34392 Montpellier Cedex 5

Abstract. A contact representation by triangles of a graph is a set of triangles

in the plane such that two triangles intersect on at most one point, each triangle

represents a vertex of the graph and two triangles intersects if and only if their

corresponding vertices are adjacent. de Fraysseix, Ossona de Mendez and Rosen-

stiehl proved that every planar graph admits a contact representation by triangles.

We strengthen this in terms of a simultaneous contact representation by triangles

of a planar map and of its dual.

A primal-dual contact representation by triangles of a planar map is a contact

representation by triangles of the primal and a contact representation by triangles

of the dual such that for every edge uv, bordering faces f and g, the intersec-

tion between the triangles corresponding to u and v is the same point as the

intersection between the triangles corresponding to f and g. We prove that every

3-connected planar map admits a primal-dual contact representation by triangles.

Moreover, the interiors of the triangles form a tiling of the triangle correspond-

ing to the outer face and each contact point is a node of exactly three triangles.

Then we show that these representations are in one-to-one correspondence with

generalized Schnyder woods defined by Felsner for 3-connected planar maps.

1 Introduction

A contact system is a set of curves (closed or not) in the plane such that two curves

cannot cross but may intersect tangentially. A contact point of a contact system is a

point that is in the intersection of at least two curves. A contact representation of a

graph G = (V,E) is a contact system C = {c(v) : v ∈ V }, such that two curves

intersect if and only if their corresponding vertices are adjacent.

The Circle Packing Theorem of Koebe [14] states that every planar graph admits a

contact representation by circles.

Theorem 1 (Koebe [14]). Every planar graph admits a contact representation

by circles.

Theorem1 implies that every planar graph has a contact representationby convexpoly-

gons, and de Fraysseix et al. [8] strengthened this by showing that every planar graph

admits a contact representation by triangles. A contact representation by triangles is

strict if each contact point is a node of exactly one triangle. de Fraysseix et al. [8]

proved the following:

?This work was partially supported by the grant ANR-09-JCJC-0041.

U. Brandes and S. Cornelsen (Eds.): GD 2010, LNCS 6502, pp. 262–273, 2011.

c ? Springer-Verlag Berlin Heidelberg 2011

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Triangle Contact Representations and Duality263

Fig.1. A strict tiling primal-dual contact representation by triangles

Theorem 2 (de Fraysseix et al. [8]). Every planar graph admits a strict contact rep-

resentation by triangles.

Moreover, de Fraysseix et al. [8] proved that strict contact representations by triangles

of a planar triangulation are in one-to-one correspondence with its Schnyder woods

defined by Schnyder [17].

Andre’ev [1] strengthen Theorem 1 in terms of a simultaneous contact representa-

tion of a planar map and of its dual. The dual of a planar map G = (V,E) is noted

G∗= (V∗,E∗). A primal-dual contact representation (V,F) of a planar map G is

two contact systems V = {c(v) : v ∈ V } and F = {c(f) : f ∈ V∗}, such that V is

a contact representation of G, and F is a contact representation of G∗, and for every

edge uv, bordering faces f and g, the intersection between c(u) and c(v) is the same

point as the intersection between c(f) and c(g). A contact point of a primal-dual con-

tact representation is a contact point of V or a contact point of F. Andre’ev [1] proved

the following:

Theorem 3 (Andre’ev [1]). Every 3-connected planar map admits a primal-dual con-

tact representation by circles.

Our main result is an analogous strengthening of Theorem 2. We say that a primal-dual

contact representation by triangles is tiling if the triangles corresponding to vertices

and those corresponding to bounded faces form a tiling of the triangle corresponding

to the outer face (see Figure 1). We say that a primal-dual contact representation by

triangles is strict if each contact point is a node of exactly three triangles corresponding

to vertices or faces (see Figure 1). We prove the following :

Theorem 4. Every 3-connected planar map admits a strict tiling primal-dual contact

representation by triangles.

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264 D. Gonc ¸alves, B. L´ evˆ eque, and A. Pinlou

In[12], Gansneret al. studyrepresentationofgraphsbytriangleswheretwo verticesare

adjacent if and only if their corresponding triangles are intersecting on a side (touch-

ing representation by triangles). Theorem 4 shows that for 3-connected planar graphs,

the incidence graph between vertices and faces admits a touching representation by

triangles.

The tools needed to prove Theorem 4 are introduced in section 2. In section 2.1, we

present a result of de Fraysseix et al. [10] concerning the stretchability of a contact sys-

tem of arcs. In section 2.2, we define (generalized)Schnyder woods and present related

results obtained by Felsner [4]. In Section 3, we define a contact system of arc, based

on a Schnyder wood, and show that this system of arc is stretchable. When stretched,

this system gives the strict tiling primal-dualcontactrepresentationby triangles.In Sec-

tion 4, we show that strict tiling primal-dual contact representations by triangles of a

planar map are in one-to-one correspondence with its Schnyder woods. In Section 5,

we define the class of planar maps admitting a Schnyder wood and thus a strict tiling

primal-dual contact representation by triangles. In Section 6, we discuss possible im-

provements of Theorem 4.

2 Tools

2.1 Stretchability

An arc is a non-closed curve. An internal point of an arc is a point of the arc distinct

fromits extremities.A contactsystem ofarcs is strict if eachcontactpointsis internalto

at most onearc. A contactsystem ofarcs is stretchableif thereexists a homeomorphism

which transforms it into a contact system whose arcs are straight line segments. An

extremal point of a contact system of arcs is a point onthe outer-boundaryof the system

and which is internal to no arc.

We define in Section 3 a contact system of arcs such that when stretched it gives a

strict tiling primal-dual contact representation by triangles. To prove that our contact

system of arcs is stretchable, we need the following theorem of de Fraysseix et al. [10].

Theorem 5 (de Fraysseix et al. [10]). A strict contact system of arcs is stretchable if

and only if each subsystem of cardinality at least two has at least three extremal points.

2.2 Schnyder Woods

The contact system of arcs defined in Section 3 is constructed from a Schnyder wood.

Schnyder woods where introduced by Schnyder [17] and then generalized by Fel-

sner [4]. Here we use the definition from [4] except if explicitly mentioned. We refer

to classic Schnyder woods defined by Schnyder [17] or generalized Schnyder woods

defined by Felsner [4] when there is a discussion comparing both.

Givena planarmapG. Letx0,x1,x2bethreedistinctverticesoccurringinclockwise

order on the outer face of G. The suspension Gσis obtained by attaching a half-edge

thatreachesintotheouterfaceto eachofthesespecialvertices.ASchnyderwood rooted

at x0, x1, x2is an orientation and coloring of the edges of Gσwith the colors 0, 1, 2

satisfying the following rules (see Figures 2 and 3):

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Triangle Contact Representations and Duality265

– Every edge e is oriented in one direction or in two opposite directions. We will

respectively say that e is uni- or bi-directed. The directions of edges are colored

such that if e is bi-directed the two directions have distinct colors.

– The half-edge at xiis directed outwards and colored i.

– Every vertex v has out-degree one in each color. The edges e0(v), e1(v), e2(v)

leaving v in colors 0, 1, 2, respectively, occur in clockwise order. Each edge en-

tering v in color i enters v in the clockwise sector from ei+1(v) to ei−1(v) (where

i + 1 and i − 1 are understood modulo 3).

– There is no interior face the boundary of which is a directed monochromaticcycle.

The differencewith the originaldefinitionof Schnyder[17] it that edges can be oriented

in two opposite directions.

A Schnyder wood of Gσdefines a labelling of the angles of Gσwhere every angle

in the clockwise sector from ei+1(v) to ei−1(v) is labeled i.

A Schnyder angle labellings of Gσis a labeling of the angles of Gσwith the labels

0, 1, 2 satisfying the following rules (see Figures 2 and 3):

– The two angles at the half-edge of the special vertex xihave labels i + 1 and i − 1

in clockwise order.

– Rule of vertices: The labels of the angles at each vertex form, in clockwise order,

a nonempty interval of 0’s, a nonempty interval of 1’s and a nonempty interval of

2’s.

– Rule offaces: Thelabelsoftheanglesat eachinteriorfaceform,inclockwiseorder,

a nonempty interval of 0’s, a nonempty interval of 1’s and a nonempty interval of

2’s. At the outer face the same is true in counterclockwise order.

Felsner [5] proved the following correspondence:

(a)

1

02

2

2

1

1

1

0

0

0

0

11

2

2

2

0

(b)

0

1

2

0

(c)

2

0

1

0

(d)

Fig.2. (a) Edge colored respectively with color 0, 1, and 2. We use distinct arrow types to

distinguish those colors. (b) Rules for Schnyder woods and angle labellings. (c) Example of

angle labelling around an uni-directed egde colored with color 0. (d) Example of angle labelling

around a bi-directed edge colored with colors 2 and 1.

Theorem 6 (Felsner [5]). Schnyder woods of Gσare in one-to-one correspondence

with Schnyder angle labellings.

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266 D. Gonc ¸alves, B. L´ evˆ eque, and A. Pinlou

0

0

0

0

1

0

1

2

0

0

1

1

2

2

1

2

2

0

0

1

1

0

0

1

2

0

1

2

2

2

2

Fig.3. A Schnyder wood with its corresponding angle labeling

3 Mixing Tools

Given a planar map G and a Schnyder wood of G rooted at x0, x1, x2we construct a

contact system of arcs A correspondingto the Schnyderwood by the following method

(see Figure 4):

Each vertex v is represented by three arcs a0(v),a1(v),a2(v), where the arc ai(v)

is colored i and represent the interval of angles labeled i of v. It may be the case that

ai(u) = ai(v) for some values of i, u and v. For every edge e of G, we choose a point

on its interior that we note p(e). There is also such a point on the half-edge leaving xi,

for i ∈ {0,1,2}. The points p(e) are the contact points of the contact system of arcs.

Actually the arcs of A are completely defined by the following subarcs : For each

angle labeled i at a vertex v in-between the edges e and e?, there is a subarc of ai(v)

going from p(e) to p(e?) along e and e?. Each contact point p(e) is the end of 4 such

subarcs.The Schnyderlabellingimplies that the three colors are representedat p(e) and

so the two subarcs with the same color are merged and form a longer arc.

One can easily see that this defines a contact system (there is no crossing arcs) of

arcs (there is no closed curve) whose contact points are the points p(e). It is also clear

that the arcs satisfy the following rules:

– For every edge e = vw uni-directed from v to w in color i: The arcs ai+1(v) and

ai−1(v) end at p(e) and the arc ai(w) goes through p(e).

– For every edge e = vw bi-directed, leaving v in color i and leaving w in color j:

Let k be such that {i,j,k} = {0,1,2}. The arcs aj(v) and ai(w) ends at p(e), and

the arcs ak(v) and ak(w) are equal and go through p(e).