Planar Graphs Have 1-string Representations

DOI 10.1007/s00454-009-9196-9

Planar Graphs Have 1-string Representations

Jérémie Chalopin·Daniel Gonçalves· Pascal Ochem

Received: 9 August 2008 / Revised: 15 May 2009 / Accepted: 18 May 2009 / Published online: 29 July 2009

© Springer Science+Business Media, LLC 2009

Abstract We prove that every planar graph is an intersection graph of strings in the plane such that any two strings intersect at most once.

Keywords Planar graphs·Strings

1 Introduction

Astringsis a curve of the plane homeomorphic to a segment. A stringshas two ends, the points ofsthat are not ends ofsareinternal pointsofs. Two stringss1ands2

crossif they have a common pointp∈s1∩s2and if going aroundp, we successively meets₁,s₂,s₁, ands₂. This means that a tangent point is not a “crossing.” In the following we consider string sets without tangent points.

In this paper, we consider intersection models for simple planar graphs (i.e., planar graphs without loops or multiple edges). A string representationof a graph G= (V , E)is a setΣof strings in the plane such that every vertexv∈V maps to a string v∈Σ and such thatuv∈Eif and only if the stringsuandvcross (at least once).

Similarly, asegment representationof a graphGis a string representation ofGin which the strings are segments.

An abstract of this paper appeared in the Proceedings of the eighteenth annual ACM–SIAM Symposium on Discrete algorithms (SODA 2007).

J. Chalopin

LIF, CNRS et Aix-Marseille Université, CMI, 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France D. Gonçalves (

!

LIRMM, CNRS et Université Montpellier 2, 161 rue Ada, 34392 Montpellier Cedex 05, France e-mail:daniel.goncalves@lirmm.fr

P. Ochem

LRI, CNRS et Université Paris-Sud, Bât 490, 91405 Orsay Cedex, France

These notions were introduced by Ehrlich et al. [3], who proved the following:

Theorem 1 [3]Planar graphs have a string representation.

In [9], Koebe proved that planar graphs are the contact graphs of disks in the plane.

Note that in this model the curves bounding two adjacent disks are tangent. However by inflating these circles we obtain string representations for planar graphs. In his PhD thesis, Scheinerman [10] conjectures a stronger result:

Conjecture 1 [10]Planar graphs have a segment representation.

Hartman et al. [8] and de Fraysseix et al. [4] proved Conjecture1 for bipartite planar graphs. Castro et al. [1] proved Conjecture1 for triangle-free planar graphs.

Recently de Fraysseix and Ossona de Mendez [6] extended this to planar graphs that have a 4-coloring in which every induced cycle of length 4 uses at most three colors.

Observe that, since parallel segments never cross, a set of parallel segments in a seg- ment representation of a graph induces a stable set of vertices. The construction in [4,8] (resp. [1]) has the nice property that there are only two (resp. three) possible slopes for the segments. So the construction induces a 2-coloring (resp. 3-coloring) ofG. Note that Castro et al. do not prove the 3-colorability of triangle-free planar graphs, they use such coloring of the graphs (by Grötzsch’s Theorem) in their con- struction. West [11] proposed a stronger version of Conjecture1in which only four slopes are allowed, thus using the fact that these graphs are 4-colorable.

Notice that two segments cross at most one point, whereas in the construction of Theorem1, strings may cross twice. Let us define a1-string representationas a string representation in which any two strings cross at most once. Thus the following theorem is a step towards Conjecture1.

Theorem 2 Planar graphs have a1-string representation.

Note that if we would allow and consider tangent points, this theorem would directly follow from Koebe’s theorem. Theorem 2 answers an open problem of de Fraysseix and Ossona de Mendez [5]. In the same article they noticed that The- orem2 implies that any planar multigraph has a string representation such that the number of crossings between two strings equals the number of edges between the two corresponding vertices.

In the next section we provide some definitions and prove that it is sufficient to prove this theorem for triangulations. Section3is devoted to the study of string rep- resentations of 4-connected triangulations. In this section we use a decomposition technique of 4-connected triangulations that is inspired on Whitney’s work [12] and that was recently used by the second author [7]. Then in Sect.4 we finally prove Theorem2for all triangulations.

2 Preliminaries

2.1 Restriction to Triangulations

Lemma 1 Every planar graph is an induced subgraph of some planar triangulation.

Proof LetGbe a planar graph embedded in the plane (i.e., a plane graph). The graph h(G)is obtained fromGby adding in every facef ofGa new vertexv_f adjacent to every vertex incident tof inG. Notice thath(G)is also a plane graph and thatGis an induced subgraph ofh(G). Moreover,h(G)is connected,h(h(G))is 2-connected, andh(h(h(G)))is a triangulation.

Note that we have to apply theh operator several times: if a facial walk goes through the same vertex several times, since multiples edges are not allowed, we

obtain a nontriangular face. !

It is clear that a 1-string representation of a triangulationT induces a 1-string rep- resentation for any of its induced subgraphs. It is thus sufficient to prove Theorem2 for triangulations.

2.2 String Representations

In a plane graphG, the unbounded face ofGis called theouter-faceand every other face ofG is aninner-face ofG. Anouter-vertex (resp.outer-edge) ofGis a ver- tex (resp. edge) ofGincident to the outer-face. The other vertices (resp. edges) of Gareinner-vertices(resp.inner-edges). The set of outer-vertices (resp. outer-edges, inner-vertices, and inner-edges) ofGis denoted byV_o(G)(resp.E_o(G),V_i(G), and E_i(G)). Anear-triangulationis a plane graph in which all the inner-faces are trian- gles. An edgeuvis achordof some near-triangulationT ifuvis an inner-edge linking two outer-vertices. From now on, we use the following notation: the strings corre- sponding to vertices of a graphGare denoted by bold letters, i.e., for anyv∈V (G), we denote its corresponding string byv. We need that in a 1-string representation of a plane graphG, each face ofGcorresponds to some topological region of the string representation.

Definition 1 Let G=(V , E) be a plane graph with a 1-string representation Σ.

Given a faceabcofG, consider a triplet(a, b, c)of its incident vertices. An (a, b, c)- regionabcis a region of the plane homeomorphic to a disk such that (see Fig.1):

• For any vertexv#=a,b, andc, we haveabc∩v= ∅(i.e.,abcintersects only with a,b,c).

• abc∩a∩b= ∅,abc∩b∩c= ∅, andabc∩c∩a= ∅(i.e.,a,b,cintersect outside abc).

• Bothabc∩bandabc∩care connected.

• The boundary ofabcsuccessively crosses (clockwise or anticlockwise)a,a,b,b, c,a,c.

Fig. 1 An(a, b, c)-regionabc

Note that according to this definition,abc∩ahas two components, and one end ofa is inabc. Note that the order in the triplet (a, b, c)matters: a regionτ of the plane cannot be an(a, b, c)-region and a(c, b, a)-region for example. A regionabc of the plane is an {a, b, c}-regionif it is either an(a, b, c)-region, an(a, c, b)-region, a(b, a, c)-region, a(b, c, a)-region, a(c, a, b)-region, or a(c, b, a)-region. When the verticesa,b, andcare not mentioned, we call such a region a face-region.

Definition 2 Astrong 1-string representation(S-representation, for short) of a near- triangulationT is a pair(Σ, R)such that:

(1) Σis a 1-string representation ofT.

(2) R is a set of disjoint face-regions such that for every inner-face abcof T, R contains an{a, b, c}-region.

A partial strong 1-string representation (PS-representation, for short) of a near- triangulationT is a triplet(Σ, R, F )in whichF ⊆E(T )and such that(Σ, R)is a strong 1-string representation ofT without the crossings corresponding to the edges ofF.

In a PS-representation(Σ, R, F )ofT, note thatΣ is a 1-string representation of T \F and that each inner-face ofT has a corresponding face-region inR.

2.3 Special Triangulations

In a near-triangulationT, aseparating 3-cycleCis a cycle of length 3 such that some vertices ofT lie insideC, whereas other vertices lie outside. It is well known that a triangulation is 4-connected if and only if it contains no separating 3-cycle. In [12], Whitney considered a special family of near-triangulations, it is why we call them W-triangulations.

Definition 3 AW-triangulation is a 2-connected near-triangulation containing no separating 3-cycle.

In particular, any 4-connected triangulation is a W-triangulation. Note that since a W-triangulation has no cut vertex, its outer-edges induce a cycle. The following lemma gives a sufficient condition for a subgraph of a W-triangulationT to be a W-triangulation.

Fig. 2 3-boundary ofT

Lemma 2 LetT be a W-triangulation and consider a cycleC ofT.The subgraph induced by the vertices lying on and insideCis a W-triangulation.

Proof Consider the near-triangulationT^& inside some cycle C of T. By definition, T has no separating 3-cycle, and consequentlyT^& does not have any separating 3- cycle. SinceT^&is clearly connected and has more than two vertices, we prove that it is 2-connected by showing that it does not contain any cut vertex.

Since the cycleC delimits the outer-face of T^&, any vertex v∈V (T^&)appears at most once on the outer face. Since the outerface appears at most once aroundv and since all its other incident faces are triangles,T^&contains a path linking all the neighbors ofv. This implies thatT^&\vis connected, and thusT^&has no cut vertex.! Definition 4 A W-triangulationT is3-bounded if the outer-boundary of T is the union of three paths(a₁, . . . , ap),(b₁, . . . , bq), and(c₁, . . . , cr)that satisfy the fol- lowing conditions (see Fig.2):

• a₁=c_r,b₁=a_p, andc₁=b_q.

• the paths are nontrivial, i.e.,p≥2,q≥2, andr≥2.

• there exists no chordaiaj,bibj, orcicj.

Such a3-boundaryofT will be denoted by(a₁, . . . , ap)–(b₁, . . . , bq)–(c₁, . . . , cr).

In the following, we will use the order on the three paths and their direc- tions, i.e.,(a₁, . . . , a_p)–(b₁, . . . , b_q)–(c₁, . . . , c_r)will be different from(b₁, . . . , b_q)–

(c₁, . . . , cr)–(a₁, . . . , ap)and(ap, . . . , a₁)–(cr, . . . , c₁)–(bq, . . . , b₁).

3 Proof for 4-connected Triangulations

The following property describes the shape of a PS-representation of a 3-bounded W-triangulation.

Property 1 Consider a 3-bounded W-triangulation T with a 3-boundary (a₁, . . . , a_p)–(b₁, . . . , b_q)–(c₁, . . . , c_r).The W-triangulationT has Property1ifT has a PS- representation(Σ, R, F )contained inside a regionτ of the plane homeomorphic to the disk that satisfies the following properties(see Fig.3):

Fig. 3 Property1

(a) F =E_o(T )\ {a₁a₂}(i.e.,the missing crossings correspond to the outer edges, excepta₁a₂).

(b) On the boundary ofτ,we successively have the ends ofa₂,a₃, . . . ,a_p,b₁, . . . , bq,c1, . . . ,cr.

If going clockwise (resp. anticlockwise) around the boundary ofτ, we cross the strings in the order described in (b), we say that the PS-representation isclockwise (resp.anticlockwise). Note that by an axial symmetry, one can obtain a clockwise PS-representation from an anticlockwise PS-representation, and vice versa. Observe that sincea_p=b₁,b_q=c₁, andc_r =a₁, both ends ofb₁andc₁lie on the boundary ofτ, but it is not the case fora₁or any other string (i.e., all the strings appearing on the boundary ofτ have an end insideτ exceptb1andc1).

Before proving that each 3-bounded W-triangulation has Property1, we give some definitions and we present Property2. Consider a 3-bounded W-triangulationT #=K₃ whose boundary is(a₁, . . . , a_p)–(b1, . . . , b_q)–(c1, . . . , c_r)and such thatT does not contain any chordaibj oraicj. LetD⊆V_i(T )be the set of inner-vertices ofT that are adjacent to some vertexa_i withi >1 (the black vertices on the left of Fig. 4).

SinceT has at least 4 vertices, no separating 3-cycle, and no chord aiaj,aibj, or a_ic_j, it follows thata₁anda₂(resp.b₁andb₂) have exactly one common neighbor inVi(T )that will be denoteda(resp.d₁).

Since there is no chorda_ia_j,a_ib_j, ora_ic_j, for each vertexa_i withi∈ [2, p−1], all the neighbors ofa_i (resp.a_p) excepta_i−1anda_i+1(resp.a_p−1andb₂) are inD.

Since for eachi∈ [2, p], there is a path linking the neighbors ofa_i inDand since the verticesa_i anda_i+1have a common neighbor inD, the setD induces a connected graph. Sinceais inD, the setD∪ {a₁}also induces a connected graph.

Definition 5 Theadjacent pathof T with respect to the 3-boundary(a₁, . . . , a_p)–

(b₁, . . . , bq)–(c₁, . . . , cr)is the shortest path linkingd₁ anda₁inT[D∪ {a₁}](the graph induced byD∪ {a₁}). This path will be denoted(d₁, d₂, . . . , d_s, a₁).

Observation 1 There exists neither an edgedidj with2≤i+1< j≤snor an edge a₁d_i with1≤i < s.Otherwise,(d₁, d₂, . . . d_s, a₁) would not be the shortest path betweend₁anda₁.

Fig. 4 The adjacent path ofT and the graphT_d2a5

Definition 6 For each edgedxay∈E(T )withx∈ [1, s]andy∈ [2, p], the graph T_dxay is the graph lying inside the cycle C =(a₁, d_s, . . . , d_x, a_y, . . . , a_p, b₂, . . . , b_q, c₂, . . . , c_r)(see Fig.4).

Note that since D⊆V_i(T ), C is a cycle, and by Lemma 2, Td_xa_y is a W- triangulation. The following property describes the shape of a PS-representation ofT_dxay.

Property 2 Consider a 3-bounded W-triangulation T with a 3-boundary (a₁, . . . , a_p)–(b₁, . . . , b_q)–(c₁, . . . , c_r)that does not have any chord a_ib_j or a_ic_j, and let (d₁, d₂, . . . , d_s, a₁)be its adjacent path.Consider an edged_xa_y∈E(T )withy >1.

The W-triangulation T_dxay has Property 2 if T_dxay has a PS-representation (Σ, R, F )satisfying the following properties (see Fig.5):

(a) F =E_o(G)\ {d_xa_y}.

(b) Every stringv∈Σ\ {d_x,a_y}is contained in a regionτ of the plane homeomor- phic to the disk. Furthermore,d_xanda_yhave their ends inτ (or on the boundary ofτ), but they cross each other outsideτ.

(c) Each face-region ofRis contained insideτ.

(d) On the boundary ofτ, we successively have the ends ofay, . . . ,ap,b1, . . . ,bq, c₁, . . . ,c_r,

a₁,d_s, . . . ,d_x+1, and then we successively have internal points of d_x,a_y,d_x, anda_y.

Here again, if going clockwise (resp. anticlockwise) around the boundary ofτ, we cross the strings in the order described in(d), we say that the PS-representation is clockwise(resp.anticlockwise). In the proof of Theorem2, we only use Property1.

However, in order to prove Property1, we use Property2. We prove these two prop- erties by doing a “crossed” induction.

Proof of Properties1and2

We prove, by induction onm≥3, that the following two statements hold:

Fig. 5 Property2

Fig. 6 Initial case for Property1

– Property1holds ifT has at mostmedges.

– Property2holds ifT_dxay has at mostmedges.

The initial case,m=3, is easy to prove since there is only one W-triangulation having at most 3 edges, K₃. For Property 1, we have to consider all the possible 3-boundaries ofK₃. All these 3-boundaries are equivalent, so letV (K₃)= {a, b, c} and consider the 3-boundary(a, b)–(b, c)–(c, a). In Fig.6there is a PS-representation (Σ, R, F )ofK₃withF = {bc, ac}that fulfills Property1. For Property 2, since a W-triangulationT_dxay has at least 4 vertices,a₁,b₁,c₁, andd₁, we haveT_dxay #=K₃, and there is no W-triangulationT_dxay with at most 3 edges. So by vacuity, Property2 holds forT_dxay with at most 3 edges.

The induction step applies to both Property1and Property2. This means that we prove Property1(resp. Property2) for the W-triangulationsT (resp.T_dxay) withm edges using both Property1 and Property2 on W-triangulations with less thanm edges. We first prove the induction for Property1.

Case 1: Proof of Property1 for a W-triangulation T withmedges Let (a₁, . . . , a_p)–(b1, . . . , b_q)–(c1, . . . , c_r)be the 3-boundary ofT considered. We distinguish dif- ferent cases according to the existence of a chordaibjoraicjinT. We successively consider the case where there is a chorda₁bi with 1< i < q, the case where there is a chorda_ib_j with 1< i < pand 1< j≤q, and the case where there is a chord a_ic_j with 1< i≤p and 1< j < r. We then finish with the case where there is no chordaibj with 1≤i≤pand 1≤j≤q(by the definition of 3-boundary,T has no chorda₁b_q,a_ib₁, ora_pb_j) and no chorda_ic_j with 1≤i≤pand 1≤j≤r(by the definition of 3-boundary,T has no chorda_pc₁,a_ic_r, ora₁c_j).

Fig. 7 Case 1.1: Chorda1bi

Fig. 8 Case 1.1:(Σ, R, F )

Case 1.1: There is a chord a₁b_i with 1 < i < q (see Fig. 7) Let T₁ (resp.

T₂) be the subgraph of T that lies inside the cycle (a₁, b_i, . . . , b_q, c₂, . . . , c_r) (resp.(a₁, a₂, . . . , a_p, b₂, . . . , b_i, a₁)). By Lemma2,T₁andT₂are W-triangulations.

SinceT has no chorda_xa_y,b_xb_y, orc_xc_y,(b_i, a₁)–(cr, . . . , c₁)–(bq, . . . , b_i)(resp.

(a₁, . . . , ap)–(b₁, . . . , bi)–(bia₁)) is a 3-boundary of T₁ (resp. T₂). Furthermore, sincea₁a₂∈/E(T₁)andc₁c₂∈/E(T₂),T₁andT₂have less edges thanT, and Prop- erty1holds forT₁andT₂with the mentioned 3-boundaries. Let(Σ₁, R₁, F₁)(resp.

(Σ₂, R₂, F₂)) be a clockwise (resp. anticlockwise) PS-representation contained in the regionτ₁(resp.τ₂) obtained forT₁ (resp.T₂) withF₁=E_o(T₁)\ {a₁bi}(resp.

F₂=E_o(T₂)\ {a₁a₂}). In Fig.8we show how to associate these two representations to obtain(Σ, R, F ), an anticlockwise PS-representation ofT contained inτ. Note that the two stringsa1(resp.bi) fromΣ₁andΣ₂have been linked.

We easily verify that(Σ, R, F )satisfies Property1:

• Σ is a string representation of T \F withF =E_o(T )\ {a₁a₂}. Indeed, since V (T₁)∪V (T₂)=V (T ) and V (T₁)∩V (T₂)= {a₁, b_i}, every vertex v∈V (T ) has exactly one string inΣ. Furthermore, since (E(T₁)\F₁)∪(E(T₂)\F₂)= E(T )\F,Σis a string representation ofT \F.

• Σ is a 1-string representation. The only edge that belongs to both T₁ andT₂ is a₁b_i. Sincea₁andb_icross each other inΣ₁(a1b_i∈/F₁) but not inΣ₂(a1b_i∈F₂), a₁andb_icross exactly once inΣ.

• (Σ, R) is “strong”: Each inner-face of T is an inner-face in T₁ or T₂, and the regionsτ₁andτ₂are disjoint (so the face-regions inτ₁are disjoint from the face- regions inτ₂).

Finally we see in Fig.8that point(b)of Property1is satisfied.

Fig. 9 Case 1.2: Chordaibj

Fig. 10 Case 1.2:(Σ, R, F )

Case 1.2: There is a chord a_ib_j with 1< i < p and 1< j ≤q (see Fig.9) If there are several chordsaibj, we consider one that maximizes j, i.e., there is no chordaibk withj < k≤q. LetT₁(resp.T₂) be the subgraph ofT that lies inside the cycle (a₁, a₂, . . . , ai, bj, . . . , bq, c₂, . . . , cr) (resp. (ai, . . . , ap, b₂, . . . , bj, ai)).

By Lemma2, T₁ andT₂ are W-triangulations. Since T has no chord a_xa_y, b_xb_y, c_xc_y, or a_ib_k withk > j,(a₁, . . . , a_i)–(ai, b_j, . . . , b_q)–(c1, . . . , c_r)(resp.(a_i, b_j)–

(b_j, . . . , b₁)–(ap, . . . , a_i)) is a 3-boundary ofT₁(resp.T₂). Furthermore, sinceb₁b₂∈/ E(T₁)anda₁a₂∈/E(T₂),T₁andT₂have less edges thanT, and Property1holds for T₁andT₂with the mentioned 3-boundaries. Let(Σ₁, R₁, F₁)(resp.(Σ₂, R₂, F₂)) be an anticlockwise (resp. clockwise) PS-representation contained in the regionτ₁(resp.

τ₂) obtained forT₁(resp.T₂) withF₁=E_o(T₁)\{a₁a₂}(resp.F₂=E_o(T₂)\{aibj}).

In Fig.10we show how to associate these two representations to obtain(Σ, R, F ), an anticlockwise PS-representation ofT contained inτ. Note that in this construction the two stringsa_i(resp.b_j) fromΣ₁andΣ₂have been linked.

As in Case 1.1, we easily verify that(Σ, R, F )satisfies Property1.

Case 1.3: There is a chorda_ic_j with1< i ≤p and 1< j < r (see Fig. 11) If there are several chordsa_ic_j, we consider one which maximizesi, i.e., there is no chorda_kc_j withi < k≤p. LetT₁(resp.T₂) be the subgraph ofT that lies inside the cycle (a₁, a₂, . . . , a_i, c_j, . . . , c_r) (resp. (c_j, a_i, . . . , a_p, b₂, . . . , b_q, c₂, . . . , c_j)).

By Lemma2, T₁ andT₂ are W-triangulations. Since T has no chord a_xa_y, b_xb_y, cxcy, or akcj withk > i,(a₁, . . . , ai)–(ai, cj)–(cj, . . . , cr)(resp.(cj, ai, . . . , ap)–

(b₁, . . . , bq)–(c₁, . . . , cj)) is a 3-boundary ofT₁(resp.T₂). Furthermore, sinceb₁b₂∈/ E(T₁)anda₁a₂∈/E(T₂),T₁andT₂have less edges thanT, and Property1holds for T₁andT₂with the mentioned 3-boundaries. Let(Σ₁, R₁, F₁)(resp.(Σ₂, R₂, F₂)) be an anticlockwise PS-representation contained in the regionτ₁(resp.τ₂) obtained for T₁(resp.T₂) withF₁=E_o(T₁)\ {a₁a₂}(resp.F₂=E_o(T₂)\ {c_ja_i}). In Fig.12we

Fig. 11 Case 1.3: Chordaicj

Fig. 12 Case 1.3:(Σ, R, F )

show how to associate these two representations to obtain(Σ, R, F ), an anticlock- wise PS-representation ofT contained inτ. Note that in this construction the two stringsa_i(resp.c_j) fromΣ₁andΣ₂have been linked.

As in Case 1.1, we easily verify that(Σ, R, F )satisfies Property1.

Case 1.4: There is no chorda_ib_j with1≤i≤pand1≤j ≤q, and no chorda_ic_j with1≤i≤pand 1≤j ≤r (see Fig.13) In this case we consider the adjacent path(d₁, . . . , ds, a₁)(see Fig.4) ofT with respect to its 3-boundary,(a₁, . . . , ap)–

(b₁, . . . , bq)–(c₁, . . . , cr). Consider the edge dsay with 1< y≤p and which min- imizesy. This edge exists since, by the definition of the adjacent path,ds is adja- cent to some vertexaywithy >1. The W-triangulationTd_sa_y having less edges than T (a1a₂∈/E(Td_sa_y)), Property2 holds forTd_sa_y. Let (Σ^&, R^&, F^&)be an anticlock- wise PS-representation almost contained in the regionτ^& obtained for T_dsay, with

F^&=E_o(T_dsay)\ {d_sa_y}.

Now we distinguish two cases according to the position of a_y: either y =2 (Case 1.4.1), ory >2 (Case 1.4.2).

Case 1.4.1:y=2 In Fig.14, starting from(Σ^&, R^&, F^&), we show how to extend the stringa1∈Σ^&(in order to crossds anda2) and how to draw the(a₁, a₂, d_s)-region a1a2dsto obtain(Σ, R, F ), an anticlockwise PS-representation ofT contained in a regionτ.

One can verify on Fig.14that(Σ, R, F )satisfies Property1.

Case 1.4.2:y >2 Let us denote bye₁, e₂, . . . , e_tthe neighbors ofd_s strictly inside the cycle(d_s, a₁, a₂, . . . , a_y, d_s), going “from right to left” (see Fig.13). By mini- mality ofywe havee_i#=a_j for all 1≤i≤tand 1≤j ≤y.

Fig. 13 Case 1.4: No chorda_ib_jora_ic_j

Fig. 14 Case 1.4.1

LetT₁be the subgraph ofT that lies inside the cycle(a₁, . . . , a_y, e₁, . . . , e_t, a₁).

By Lemma2,T₁is a W-triangulation. Since the W-triangulationT has no separating 3-cycle(d_s, a₁, e_i),(d_s, a_y, e_i), or(d_s, e_i, e_j), there exists no chorda₁e_i,a_ye_i, ore_ie_j inT₁. So (a₂, a₁)–(a1, e_t, . . . , e₁, a_y)–(ay, . . . , a₂) is a 3-boundary of T₁. Finally, sinceT₁has less edges thanT (a1d_s∈/E(T₁)), Property1holds forT₁with respect to the mentioned 3-boundary. Let(Σ₁, R₁, F₁)be a clockwise PS-representation con- tained in the regionτ₁obtained forT₁withF₁=E₀(T₁)\ {a₂a₁}.

In Fig.15, starting from(Σ^&, R^&, F^&)and(Σ₁, R₁, F₁), we show how to join the stringsa1(resp.ay) ofΣ^&andΣ₁, how to extend the stringseifor 1≤i≤t, and how to draw the face-regionsaye1ds,eta1ds, andeiei−1dsfor 2≤i≤t, in order to obtain (Σ, R, F ), an anticlockwise PS-representation ofT contained in a regionτ.

We verify that(Σ, R, F )satisfies Property1:

• Σ is a string representation of T \F withF =E_o(T )\ {a₁a₂}. Indeed, since V (T_dsay)∪V (T₁)=V (T ) and V (T_dsay)∩V (T₁)= {a₁, a_y}, every vertex v∈ V (T )has exactly one string in Σ. Furthermore, since E(T )\F =(E(T_dsay)\

F^&)∪(E(T₁)\F₁)∪ {a_ye₁, e_ta₁, d_sa₁} ∪{e_ie_i−1|i∈ [2, t]} ∪ {d_se_i|i∈ [1, t]},Σ

is a string representation ofT \F.

Fig. 15 Case 1.4.2

• Σ is a 1-string representation. Indeed Td_sa_y andT₁ do not have common edges, and the new crossings added correspond to edges missing in bothE(Td_sa_y)\F^&

andE(T₁)\F₁.

• (Σ, R)is “strong”: The only inner-faces ofT not inT_dsay nor inT₁are the faces d_sa_ye₁,d_sa₁e_t, andd_se_ie_i+1with 1≤i < t. These faces correspond to the new face-regions.

Finally we see in Fig.15that point(b)of Property1is satisfied.

So Property1holds for any W-triangulationT withmedges, and this concludes the proof of Case 1.

Case 2: Proof of Property2for a W-triangulationT_dxay withmedges Recall that the W-triangulation Td_xa_y is a subgraph of a W-triangulation T with 3-boundary (a₁, . . . , ap)–(b₁, . . . , bq)–(c₁, . . . , cr). Moreover,T has no chordaibj oraicj, and its adjacent path is (d₁, . . . , d_s, a₁), with s ≥1. We distinguish the case where d_xa_y=d₁a_pand the case whered_xa_y#=d₁a_p.

Case 2.1: d_xa_y=d₁a_p (see Fig. 16) Let T₁ be the subgraph of T_d1ap that lies inside the cycle (a₁, d_s, . . . , d₁, b₂, . . . , b_q, c₂, . . . , c_r). By Lemma 2, T₁ is a W- triangulation. This W-triangulation has no chordb_ib_j,c_ic_j,d_id_j, ora₁d_j. We con- sider two cases according to the existence of an edged₁bi with 2< i≤q.

• IfT₁has no chordd₁b_i, then(d₁, b₂, . . . , b_q)–(c1, . . . , c_r)–(a1, d_s, . . . , d₁)is a 3- boundary ofT₁.

• IfT₁ has a chordd₁bi with 2< i≤q, note thatq >2 and that there cannot be a chordb₂a₁ or b₂d_j with 1< j ≤s (this would violate the planarity ofT_dxay, see Fig.16). So in this case, (b₂, d₁, . . . , d_s, a₁)–(cr, . . . , c₁)–(bq, . . . , b₂)is a 3- boundary ofT₁.

Finally, since T₁ is a W-triangulation with less edges than T_d1ap (b₁b₂∈/E(T₁)), Property1holds forT₁with respect to at least one of the two mentioned 3-boundaries.

Fig. 16 Case 2.1:

Td_xa_y=Td₁a_p

Fig. 17 Case 2.1:(Σ, R, F )

Whichever 3-boundary we consider, we obtain a PS-representation(Σ₁, R₁, F₁)of T₁contained in a regionτ₁, with the same following characteristics:

• F₁=E_o(T )\ {d₁b₂},

• in the boundary ofτ₁, we successively meet the ends ofd₁, . . . ,d_s,a₁,c_r, . . . ,c₁, bq, . . . ,b2(clockwise or anticlockwise).

In Fig. 17 we modify (Σ₁, R₁, F₁), by extending the strings d₁ and b₂ and by adding a new stringa_p and a new face-region d₁b₂a_p. This leads to(Σ, R, F ), a PS-representation ofT_d1apcontained in a regionτ.

We verify that(Σ, R, F )satisfies Property2:

• Σ is a 1-string representation of Td₁a_p\F: Indeed,E(Td₁a_p)\F is the disjoint union ofE(T₁)\F₁and{a_pd₁}.

• (Σ, R)is “strong”: The only inner-face ofT_d1apthat is not an inner-face ofT₁is d₁apb₂, which corresponds to the new face-regiond₁apb₂.

Finally we see in Fig.17that the other points of Property2are satisfied.

Case 2.2:T_dxay #=T_d1ap In this case we consider an edge d_za_w∈E(T_dxay)such thatdzaw#=dxay. Among all the possible edgesdzaw, we choose the one that first maximizeszand then minimizesw. Such an edge necessarily exists, and actually one can see thatd_z=d_xord_z=d_x−1. Indeed, ifd_x=d₁, there is at least one edged₁a_w withw > y, the edged₁ap. Ifx >1, it is clear by the definition of the adjacent path that the vertexd_x−1is adjacent to at least one vertexa_w withw≥y.

Fig. 18 Case 2.2.1:z=xandw=y+1

Fig. 19 Case 2.2.1:(Σ, R, F )

By Lemma 2, T_dzaw is a W-triangulation. Since d_xa_y ∈/ E(T_dzaw), the W- triangulationTd_za_whas less edges thanTd_xa_y, and so Property2holds forTd_za_w. Let

(Σ^&, R^&, F^&)be an anticlockwise PS-representation almost contained in the regionτ^&

obtained forT_dzaw withF^&=E_o(T_dzaw)\ {d_za_w}.

We distinguish four cases according to the edgedzaw. Whenz=x, we consider the case wherew=y+1 and the case wherew > y+1. Whenz=x−1, we consider the case wherew=yand the case wherew > y.

Case 2.2.1:T_dxay #=T_d1ap,z=x, andw=y+1(see Fig.18) In Fig.19we mod-

ify(Σ^&, R^&, F^&)by adding a new stringayand a new face-regionayawdx. This leads

to(Σ, R, F ), an anticlockwise PS-representation ofTd_xa_y almost contained in a re- gionτ.

We verify that(Σ, R, F )satisfies Property2:

• Σ is a 1-string representation ofTd_xa_y \F: Indeed,E(Td_xa_y)\F is the disjoint union ofE(T_dzaw)\F^&and{d_xa_y}.

• (Σ, R)is “strong”: The only inner-face ofT_dxay that is not an inner-face ofT_dzaw isdxayaw, which corresponds to the new face-regiondxayaw.

Finally we see in Fig.19that the other points of Property2are satisfied.

Fig. 20 Case 2.2.2:

Td_xa_y#=Td₁a_p,z=x−1, and w=y

Fig. 21 Case 2.2.2:(Σ, R, F )

Case 2.2.2:z=x−1andw=y(see Fig.20) In Fig.21, we modify(Σ^&, R^&, F^&) by extending the string dx and by adding a new face-regiondxdzay. This leads to (Σ, R, F ), an anticlockwise PS-representation of T_dxay almost contained in a re- gionτ.

We verify that(Σ, R, F )satisfies Property2:

• Σ is a 1-string representation ofT_dxay \F: Indeed,E(T_dxay)\F is the disjoint union ofE(T_dzaw)\F^&and{d_xd_z, d_xa_y}.

• (Σ, R)is “strong”: The only inner-face ofTd_xa_y that is not an inner-face ofTd_za_w

isdxdzay, which corresponds to the new face-regiondxdzay.

Finally we see in Fig.21that the other points of Property2are satisfied.

Case 2.2.3:z=x andw > y+1 (see Fig.22) Let us denote bye₁, e₂, . . . , et the neighbors ofdx strictly inside the cycle (dx, ay, . . . , aw, dx), going “from right to left” (see Fig. 22). Since there is no chord aiaj, we have t≥1. Furthermore by minimality of w we havee_i #=a_j for all 1≤i≤t andy≤j ≤w. Let T₁ be the subgraph ofT_dxay that lies inside the cycle(a_y, . . . , a_w, e₁, . . . , e_t, a_y). By Lemma2, T₁ is a W-triangulation. Since the W-triangulationT_dxay has no separating 3-cycle (d_x, a_w, e_i)or(d_x, e_i, e_j), there exists no chorda_we_i ore_ie_j inT₁. With the fact that

Fig. 22 Case 2.2.3:

Td_xa_y#=Td₁a_p,z=x, and w > y+1

Fig. 23 Case 2.2.3:(Σ, R, F )

t≥1, we know that (e_t, a_y)–(ay, . . . , a_w)–(aw, e₁, . . . , e_t)is a 3-boundary of T₁. Finally, sinceT₁has less edges thanT_dxay (dxa_y∈/E(T₁)), Property 1 holds forT₁ with respect to the mentioned 3-boundary. Let(Σ₁, R₁, F₁)be an anticlockwise PS- representation contained in the regionτ₁obtained forT₁withF₁=E₀(T₁)\ {etay}.

In Fig.23, starting from(Σ^&, R^&, F^&)and(Σ₁, R₁, F₁), we show how to join the stringsawofΣ^&andΣ₁, how to extend the stringayand the stringseifor 1≤i≤t, and how to draw the face-regionsayetdx, e1awdx, and eiei−1dx for 1< i≤t, in order to obtain(Σ, R, F ), an anticlockwise PS-representation ofTd_xa_y contained in a regionτ.

We verify that(Σ, R, F )satisfies Property2:

• Σ is a 1-string representation ofT_dxay \F withF =E_o(T_dxay)\ {d_xa_y}: Indeed, E(T_dxay)\F is the disjoint union ofE(T_dzaw)\F^&,E(T₁)\F₁, and{a_we₁, d_xa_y}∪

{e_ie_i−1|i∈ [2, t]} ∪ {d_xe_i |i∈ [1, t]}.

• (Σ, R)is “strong”: The only inner-faces ofT_dxay that are not inner-faces inT_dzaw orT₁ared_xa_ye_t,d_xa_we₁, and the facesd_xe_ie_i−1for 2≤i≤t, which correspond to the new face-regions.

Finally we see in Fig.23that the other points of Property2are satisfied.

Fig. 24 Case 2.2.4:T_dxay#=T_d1ap,z=x−1, andw > y

Fig. 25 Case 2.2.4:(Σ, R, F )

Case 2.2.4:z=x−1 andw > y(see Fig.24) Let us denote bye₁, e₂, . . . , e_t the neighbors ofd_zstrictly inside the cycle(d_z, d_x, a_y, . . . , a_w, d_z), going “from right to left” (see Fig.24). By maximality ofz, there is no edged_xa_w, sot≥1. Let us denote byf₁, . . . , f_u the neighbors of d_x strictly inside the cycle(d_x, a_y, . . . , a_w, d_z, d_x), going “from right to left” (see Fig.24). Note that f₁=e_t and that by minimality ofw, there is no edged_za_y, sou≥1.

By minimality ofw (resp. maximality ofz) we havee_i#=a_j (resp.f_i #=a_j) for all 1≤i≤t (resp. 1≤i≤u) andy≤j ≤w. LetT₁be the subgraph ofT_dxay that lies inside the cycle(ay, . . . , aw, e₁, . . . , et, f₂, . . . , fu, ay). By Lemma2,T₁is a W- triangulation. Since the W-triangulationTd_xa_y has no separating 3-cycle(dz, aw, ei), (dz, ei, ej),(dx, fi, fj), or (dx, fi, ay), there exists no chord awei, eiej, fifj, or f_ia_y inT₁. With the fact thatt ≥1 andu≥1, we know that (f₁, f₂, . . . , f_u, a_y)–

(a_y, . . . , a_w)–(aw, e₁, . . . , e_t)is a 3-boundary ofT₁. Finally, sinceT₁has less edges thanT_dxay (dxa_y∈/E(T₁)), Property1 holds for T₁ with respect to the mentioned 3-boundary. Let(Σ₁, R₁, F₁)be an anticlockwise PS-representation contained in the regionτ₁obtained forT₁withF₁=E₀(T₁)\ {f₁f₂}.

Fig. 26 S-representation ofT from(Σ, R, F )

In Fig.25, starting from(Σ^&, R^&, F^&)and(Σ₁, R₁, F₁), we show how to join the stringsa_wofΣ^&andΣ₁, how to extend the stringd_x,a_y, the stringse_ifor 1≤i≤t, and the stringsf_i for 2≤i≤u, and how to draw the face-regionsd_za_we₁,d_ze_ie_i−1 for 2≤i≤t,dzdxet,dxfifi−1for 2≤i≤u, anddxayfuin order to obtain(Σ, R, F ), an anticlockwise PS-representation ofT_dxay almost contained in a regionτ.

We verify that(Σ, R, F )satisfies Property2:

• Σ is a 1-string representation of T_dxay \F with F =E_o(T_dxay)\ {d_xa_y}: In- deed, E(T_dxay)\F is the disjoint union of E(T_dzaw)\ F^&, E(T₁)\F₁, and {d_xa_y, d_xd_z, a_we₁, a_yf_u} ∪{d_ze_i |i∈ [1, t]} ∪ {d_xf_i |i∈ [1, u]} ∪ {e_ie_i−1|i∈ [2, t]} ∪ {fifi−1|i∈ [2, u]}.

• (Σ, R)is “strong”: The only inner-faces ofTd_xa_y that are not inner-faces inTd_za_w

or T₁ are d_za_we₁, d_ze_ie_i−1 for 2≤i≤t, d_zd_xe_t,d_xf_if_i−1 for 2≤i≤u, and d_xa_yf_u, which correspond to the new face-regions.

Finally we see in Fig.25that the other points of Property2are satisfied. So, Prop- erty2holds for any W-triangulationT_dxaywithmedges, and this completes the proofs of Properties1and2.

4 Proof in the General Case

Theorem 3 Every triangulationT admits an S-representation(Σ, R).

Proof We prove this result by induction on the number of separating 3-cycles. Note that any triangulationT is 3-connected and that ifT has no separating 3-cycle, then T is 4-connected and is a W-triangulation. Consequently, ifT is a 4-connected trian- gulation whose outer-vertices area,b, andc, thenT is a W-triangulation 3-bounded by(a, b)–(b, c)–(c, a). By Property1,T admits a PS-representation(Σ, R, F ), with F = {bc, ca}, that is contained in a regionτ. Furthermore, in the boundary ofτ, we successively meet the ends ofb,b,c,c,a. To obtain an S-representation of T, it is sufficient to extenda,b, andcoutside ofτ so thatccrossesaandb, as depicted in Fig.26.

Suppose now thatT is a triangulation that contains at least one separating 3-cycle.

Consider a separating 3-cycle(a, b, c)such that there is no other separating 3-cycle lying inside. This implies that the triangulationT^& induced by the vertices on and inside(a, b, c)is 4-connected.

LetT₁be the triangulation obtained by removing the vertices lying strictly inside (a, b, c). LetT₂ be the subgraph of T induced by the vertices lying strictly inside