Algorithms for embedded graphs

Transcription

1 Algorithms for embedded grphs Éric Colin de Verdière October 4, 2017

2 ALGORITHMS FOR EMBEDDED GRAPHS Foreword nd introduction Foreword This document is the overlpping union of some course notes tht the uthor used in previous yers for grdute courses. It is certinly not in nl shpe, nd comments by e-mil re welcome. Ech exercise is lbeled with one to three strs, supposed to be n indiction of its importnce (in prticulr, depending on whether the result or technique is used lter), not of its diculty. Introduction This is prtil introduction to the computtionl spects of grphs drwn without crossings in the plne or in more complicted surfces. This topic hs been subject of ctive reserch, especilly over the lst decde, nd is relted to rther diverse elds nd communities: in grph lgorithms: As we shll see, becuse plnr grphs ber importnt properties, mny generl grph problems become esier when restricted to plnr grphs (shortest pth, ow nd cut, minimum spnning trees, vertex cover, grph isomorphism, etc.). The sme holds for grphs on surfces, to some extent; in grph theory, the theory of grph minors founded by Robertson nd Seymour mkes hevy use of grphs embeddble on xed surfce, s well s grphs excluding xed minor. Edge-width nd fce- Dte of this version: October 4, Ltest version vilble t univ-mlv.fr/~colinde/cours/ll-lgo-embedded-grphs.pdf. Foreword nd introduction width re closely relted to the notion of shortest non-contrctible closed curve; in topology, the clssiction of surfces, s discovered in the beginning of the 20th century, is inherently lgorithmic. Surfces ply prominent role in the deep theories of knots nd three-mnifolds; there re lso mny lgorithmic questions in these res; in computtionl geometry, surfces rise nturlly in vrious pplictions. Opertions in geometric spces such s decomposition, extrction of importnt fetures, nd shortest pth computtion re bsic computtionl geometry tsks tht re relevnt in prticulr for surfces, usully embedded in R 3, or even plnr surfces. Mny grphs encountered in prctice re geometric, nd either re plnr or hve few crossings (think of rod network with few overpsses nd underpsses). Thus it mkes sense to look for ecient lgorithms dedicted to such grphs. In ddition, in computer grphics, one needs to eciently process surfces represented by tringulr meshes, e.g., to cut them to mke them plnr; we shll introduce lgorithms for such purposes. Outline The rst chpter introduces plnr grphs from the topologicl nd combintoril point of view. The second chpter considers the problem of testing whether grph is plnr, nd, if so, of drwing it without crossings in the plne. Then we move on with some generl grph problems, for which we give ecient lgorithms when the input grph is plnr. Then, we consider grphs on surfces (plnr grphs being n importnt specil cse). In Chpter 4, we introduce surfces from the topologicl point of view; in Chpter 5, we present lgorithms using the cut locus to build short curves nd decompositions of surfces. In Chpter 6, we introduce two importnt topologicl concepts, homotopy nd the universl cover. All these techniques re combined in Chpter 7 to provide lgorithms to shorten curves up to deformtion. 2

3 ALGORITHMS FOR EMBEDDED GRAPHS 1. Bsic properties of plnr grphs Some prts of these notes re not used in lter sections, nd cn be sfely skipped by the reder not interested in them. Such optionl prts re Chpter 2, nd Chpter 3 except Section 3.4 (which serves s gentle introduction to the concept of cross-metric surfce). Only prt of the mteril covered in this course ppered in textbooks. For gentle introductions to topology, see Armstrong [3] nd Stillwell [78]. Mny plnr grph lgorithms re treted in the course notes by Klein nd Mozes [55]. For grphs on surfces from combintoril point of view, see Mohr nd Thomssen [68]. A broder overview on computtionl topology of grphs on surfces is given in Colin de Verdière [17]. For wider perspective in generl computtionl topology, see the recent course notes by Erickson [31]. Acknowledgments I would like to thnk severl people who suggested some corrections in previous versions or provided vluble informtion: Je Erickson, Éric Fusy, Frncis Lzrus, Arnud de Mesmy, Arthur Milchior, nd Vincent Pilud. Chpter 1 Bsic properties of plnr grphs 1.1 Topology Preliminries on topology We ssume some fmilirity with bsic topology, but we recll some denitions nonetheless. A topologicl spce is set X with collection of subsets of X, clled open sets, stisfying the three following xioms: the empty set nd X re open; ny union of open sets is open; ny nite intersection of open sets is open. There is, in prticulr, no notion of metric (or distnce, ngle, re) in topologicl spce. The open sets give merely n informtion of neighborhood: neighborhood of x X is set contining n open set contining x. This is lredy lot of informtion, llowing to dene continuity, homeomorphisms, connectivity, boundry, isolted points, dimension.... Speciclly, mp f : X Y is continuous if the inverse imge of ny open set by f is n open set. If X nd Y re two topologicl spces, mp f : X Y is homeomorphism if it is continuous, bijective, nd if its inverse f 1 is lso continuous. A point of detil, ruling out pthologicl spces: the topologicl spces considered in these notes re ssumed to be 3

4 ALGORITHMS FOR EMBEDDED GRAPHS 1. Bsic properties of plnr grphs Figure 1.1. The stereogrphic projection. Husdor, which mens tht two distinct points hve disjoint neighborhoods. Exmple 1.1. Most of the topologicl spces here re endowed with nturl metric, which should be forgotten, but dene the topology: R n (n 1); the n-dimensionl sphere S n, i.e., the set of unit vectors of R n+1 ; the n-dimensionl bll B n, i.e., the set of vectors in R n of norm t most 1; in prticulr B 1 = [ 1, 1] nd [0, 1] re homeomorphic; the set of lines in R 2, or more generlly the Grssmnnin, the set of k-dimensionl vector spces in R n. Exercise 1.2 (stereogrphic projection). 99 Prove tht the plne is homeomorphic to S 2 with n rbitrry point removed. (Indiction: see Figure 1.1.) A closed set in X is the complement of n open set. The closure of subset of X is the (unique) smllest closed set contining it. The interior of subset of X is the (unique) lrgest open set contined in it. The boundry of subset of X equls its closure minus its interior. A topologicl spce X is compct if ny set of open sets whose union is X dmits nite subset whose union is still X. A pth in X is continuous mp p : [0, 1] X; its endpoints re p(0) nd p(1). Its reltive interior is the imge by p of the open intervl (0, 1). A pth is simple if it is one-to-one. A spce X is connected 1 if it is nonempty nd, for ny points nd b in X, there exists pth in X whose endpoints re nd b. The connected components of topologicl spce X re the clsses of the equivlence reltion on X dened by: is equivlent to b if there exists pth between nd b. A topologicl spce X is disconnected (or seprted) by Y X if nd only if X \ Y is not connected; points in dierent connected components of X \ Y re seprted by Y Grphs nd embeddings We will use stndrd terminology for grphs. Unless noted otherwise, ll grphs re undirected nd nite but my hve loops nd multiple edges. A circuit in grph G is closed wlk without repeted vertices. 2 A grph yields nturlly topologicl spce: for ech edge e, let X e be topologicl spce homeomorphic to [0, 1]; let X be the disjoint union of the X e ; for e, e, identify (quotient topology), in X, endpoints of X e nd X e if these endpoints correspond to the sme vertex in G. An embedding of G in the plne R 2 is continuous one-to-one mp from G (viewed s topologicl spce) to R 2. Sid dierently, it is crossing-free drwing of G on R 2, being the dt of two mps: Γ V, which ssocites to ech vertex of G point of R 2 ; Γ E, which ssocites to ech edge e of G pth in R 2 between the imges by Γ V of the endpoints of e, in such wy tht: 1 In this course, the only type of connectivity considered is pth connectivity. 2 This is often clled cycle; however, in the context of these notes, this word is lso used to men homology cycle or closed curve, so it seems preferble to void overloding it gin. 4

5 ALGORITHMS FOR EMBEDDED GRAPHS 1. Bsic properties of plnr grphs the mp Γ V is one-to-one (two distinct vertices re sent to distinct points of R 2 ); for ech edge e, the reltive interior of Γ E (e) is one-to-one (the imge of n edge is simple pth, except possibly t its endpoints); for ll distinct edges e nd e, the reltive interiors of Γ E (e) nd Γ E (e ) re disjoint (two edges cnnot cross); for ech edge e nd for ech vertex v, the reltive interior of Γ E (e) does not meet Γ V (v) (no edge psses through vertex). We cn ctully replce R 2 bove with ny topologicl spce Y nd tlk bout n embedding of grph in Y. When we spek of embedded grphs, we sometimes implicitly identify the grph, its embedding, nd the imge of its embedding Plnr grphs nd the Jordn curve theorem In the remining prt of this chpter, we only consider embeddings of grphs into the sphere S 2 or the plne R 2. A grph is plnr if it dmits n embedding into the plne. By Exercise 1.2, this is equivlent to the existence of n embedding into the sphere S 2. The fces of grph embedding re the connected components of the complement of the imge of the vertices nd edges of the grph. Here re the most-often used results in the re. Theorem 1.3 (Jordn curve theorem, reformulted; see [80]). Let G be grph embedded on S 2 (or R 2 ). Then G disconnects S 2 if nd only if it contins circuit. Theorem 1.4 (JordnSchönies theorem; see [80]). Let f : S 1 S 2 be one-to-one continuous mp. Then S 2 \ f(s 1 ) is homeomorphic to two disjoint copies of the open disk. Exercise Sketch proof of the Jordn curve theorem in the cse where G is embedded in the plne with polygonl edges. These results re, perhps surprisingly, dicult to prove: the diculty comes from the generlity of the hypotheses (only continuity is required). For exmple, if in the Jordn curve theorem one ssumes tht G is embedded in the plne with polygonl edges, the theorem is not hrd to prove. A grph is cellulrly embedded if its fces re (homeomorphic to) open disks. Henceforth, we only consider cellulr embeddings. It turns out tht grph embedded on the sphere is cellulrly embedded if nd only if it is connected Combintorics So fr, we hve considered curves nd grph embeddings in the plne tht re rther generl Combintoril mps for plnr grph embeddings We now focus on the combintoril properties of cellulr grph embeddings in the sphere. Since we re not interested in the geometric properties, it suces to specify how the fces re glued together, or lterntively the cyclic order of the edges round vertex. Embeddings of grphs on the plne re treted similrly: just choose distinguished fce of the embedding into S 2, representing the innite fce of the embedding in the plne. An lgorithmiclly sound wy of representing combintorilly cellulr grph embedding in S 2 is vi combintoril mps. The combintoril mp ssocited to cellulr grph embedding G is the set of closed wlks in G, obtined from wlking round the boundry of ech fce of G. (In generl, these wlks my repet edges nd/or vertices). This informtion is enough to reconstruct the sphere combintorilly, by tking the bstrct 3 Although this sttement should be intuitively cler, it is not so obvious to prove. It my help to use the results of Chpter 4, especilly the fct tht every fce of grph embedding is surfce with boundry. 5

6 ALGORITHMS FOR EMBEDDED GRAPHS 1. Bsic properties of plnr grphs vi ei int vertex_degree(flg fl) { int j=0; Flg fl2=fl; do { ++j; fl2=fl2->ei()->fi(); } while (fl2!=fl); return j; } int fce_degree(flg fl) { int j=0; Flg fl2=fl; do { ++j; fl2=fl2->ei()->vi(); } while (fl2!=fl); return j; } Figure 1.3. C++ code for degree computtion. Figure 1.2. The gs re represented s line segments prllel to the edges; there re four gs per edge. The involutions vi, ei, nd on the thick g re lso shown. grph nd ttching disk to ech fcil wlk. By extension, if G is n embedding of non-connected grph, the combintoril mp of G is the disjoint union of the combintoril mps ssocited to its connected components (which re, independently, cellulrly embedded). However, in terms of dt structures, these fcil wlks re not very esy to mnipulte, so we now present more elborted dt structure tht contins the sme informtion but is more convenient. The bsic notion is the g, which represents n incidence between vertex, n edge, nd fce of the embedding. Three involutions llow to move to nerby g, nd, by iterting, to visit the whole grph embedding; see Figure 1.2: vi moves to the g with the sme edge-fce incidence, but with dierent vertex incidence; ei moves to the g with the sme vertex-fce incidence, but with dierent edge incidence; moves to the g with the sme vertex-edge incidence, but with dierent fce incidence. Exmple 1.6. Figure 1.3, left, presents code to compute the degree of vertex, i.e., the number of vertex-edge incidences of this vertex. The function tkes s input g incident with tht vertex. Note tht loop incident with the vertex mkes contribution of two to the degree. Dully, on the right, code to compute the degree of fce (the number of edge-fce incidences of this fce) is shown. Note tht g is not necessrily uniquely dened by its triple (vertex, edge, nd fce), s shows the exmple of grph with single vertex nd single (loop) edge. The complexity of grph G = (V, E) is V + E. The complexity of cellulr grph embedding is the totl number of gs involved, which is liner in the number of edges (every edge bers four gs), nd lso in the number of vertices, edges, nd fces. Therefore the complexity of grph cellulrly embedded in the plne nd of one of its embeddings re linerly relted. The dt structure indicted bove llows to nvigte throughout the dt structure, but does not store vertices, edges, nd fces explicitly. In mny cses, however, it is necessry to hve one dt structure (object) per vertex, edge, or fce. For exmple: if one hs to be ble to check in constnt time whether n edge is loop (incident twice to the sme vertex), the dt structure given bove is not sucient. On the other hnd, if every g hs pointer to the incident vertex, then testing whether n edge is loop cn be done by testing the equlity of two pointers in constnt time; 6

7 ALGORITHMS FOR EMBEDDED GRAPHS 1. Bsic properties of plnr grphs in coloring problems, one need to store colors on the vertices of the grph. Such informtion cn be stored in the dt structure used for ech vertex. For such purposes, ech g cn hve pointer to the underlying vertex, edge, nd fce (clled respectively vu, eu, fu). Ech such vertex, edge, or fce contins no informtion on the incident elements, only informtion needed in the lgorithms. If needed, one my similrly put some informtion in the vertex-edge, edge-fce, vertex-fce, nd vertex-edge-fce incidences. Mintining such informtions, however, comes with cost, which is not lwys desirble. For exmple, ssume we wnt to be ble to remove one edge incident to two dierent fces in constnt time. If we keep the informtion fu, this must tke time proportionl to the smller degree of the two fces (since the two fces re merged, the fu pointer hs to be updted t lest on one side of the edge). If we only keep vu, sy, then such n updte is not needed, nd this edge removl cn be done in constnt time Dulity nd Euler's formul A dul grph of cellulr grph embedding G = (V, E) on S 2 is grph embedding G dened s follows: put one vertex f of G in the interior of ech fce f of G; for ech edge e of G, crete n edge e in G crossing e nd no other edge of G (if e seprtes fces f 1 nd f 2, then e connects f1 nd f 2 ). See Figure 1.4. A dul grph embedding is lso cellulr. The combintoril mp of the dul grph is unique. Actully, with the mp representtion, dulizing is esy: simply replce with vi nd vice-vers. This in prticulr proves tht dulity is n involution: G = G. Exercise 1.7 (esy). 999 Every tree (cyclic connected grph) with v vertices nd e edges stises v e = 1. Lemm 1.8. Let G = (V, E) be cellulr grph embedding in S 2, nd let G = (F, E ) be its dul grph. Furthermore, let E E. Then (V, E ) is cyclic if nd only if (F, (E \ E ) ) is connected. In Figure 1.4. Dulity. prticulr, (V, E ) is spnning tree if nd only if (F, (E \ E ) ) is spnning tree. Proof. (V, E ) is cyclic if nd only if S 2 \ E is connected, by the Jordn curve theorem 1.3. Furthermore, S 2 \E is connected if nd only if (F, (E\ E ) ) is connected: Two points x nd x in fces f nd f of G cn be connected by pth voiding E nd not entering ny fce other thn f nd f if nd only if f nd f re djcent by some edge not in E, i.e. if nd only if f nd f re djcent in (F, (E \ E ) ). Corollry 1.9 (Euler's formul for cellulr grph embeddings in S 2 ). For every cellulr grph embedding in S 2 with v vertices, e edges, nd f fces, we hve v e + f = 2. Hence this formul lso holds for every embedding of connected grph in the plne. Proof. Let T be the edge set of spnning tree of G. The dul edges of its complement, (E \ T ), is lso spnning tree. The number of edges of G is e = T + (E \ T ), which, by Exercise 1.7, equls (v 1) + (f 1). 7

8 ALGORITHMS FOR EMBEDDED GRAPHS 1. Bsic properties of plnr grphs Figure 1.5. Figure 1.4. The brycentric subdivision of the prt of the grph shown in of them 4. We will see lter tht (most of) these dt structures generlize to grphs embedded on surfces. There re further generliztions to higher dimensions [6, 61, 62]; this is importnt especilly in geometric modelling. Eppstein provides mny proofs of Euler's formul 5. Exercise 1.10 shows tht K 5 nd K 3,3 re not plnr. There is converse sttement: Kurtowski's theorem sserts tht grph G is plnr if nd only if it does not contin K 5 or K 3,3 s subdivision; in other words, if nd only if one cnnot obtin K 5 or K 3,3 from G by removing edges nd isolted vertices nd replcing every degree-two vertex nd its two incident edges with single edge [56,64,79]. Let G be cellulr embedding of grph on S 2. By overlying G with its dul grph G, we obtin qudrngultion: cellulr embedding of grph G + where ech fce hs degree four. See Figure 1.4. Every fce of G + is incident with four vertices: one vertex v G of G, one vertex v G of G, nd two vertices tht re the intersection of n edge of G nd n edge of G. If, within ech fce, we connect v G with v G, we obtin tringultion, clled the brycentric subdivision of G (Figure 1.5). Ech tringle in the brycentric subdivision corresponds to g; its three neighbors re the gs rechble vi the opertions vi, ei, nd. Exercise 1.10 (esy direction of Kurtowski's theorem). 999 Show tht the complete grph with 5 vertices, K 5, is not plnr. Indiction: Use Euler's formul nd double-counting on the number of vertex-edge nd edge-fce incidences. Also show tht the biprtite grph K 3,3 (with 6 vertices v 1, v 2, v 3, w 1, w 2, w 3 nd 9 edges, connecting every possible pir {v i, w j }) is not plnr. 1.3 Notes For more informtion on bsic topology, see for exmple Armstrong [3] or Henle [46]; see lso Stillwell [78]. For more informtions on plnr grphs, see (the next two chpters nd) Mohr nd Thomssen [68, Chpter 2]. There re mny essentilly equivlent wys of representing plnr grph embeddings [28, 54]; the computtionl geometry librry CGAL implements one 4 Chpter_min.html

9 ALGORITHMS FOR EMBEDDED GRAPHS 2. Plnrity testing nd grph drwing Chpter 2 Plnrity testing nd grph drwing Given grph G in usul form, e.g., where ech vertex hs linked list of pointers to its incident edges, nd ech edge hs two pointers to its incident vertices, how cn we determine whether G is plnr? Section 2.1 nswers this question. Then we move on by considering lgorithms to drw plnr grph in the plne or in R Plnrity testing Given grph G, how hrd is it to determine whether G is plnr? Theorem 2.1. Given grph G, one cn, in (optiml) liner time, determine whether G is plnr, nd if so, compute combintoril mp of G in the plne. We shll here prove this theorem with weker, cubic complexity. With much cre, rening these ides indeed leds to liner-time lgorithm [50]. A grph G is biconnected if it hs t lest three vertices, nd removing zero or one vertex (together with their incident edges) from G does not disconnect G. A cutvertex of G is vertex whose removl increses the number of connected components of G. A block of G is n inclusionwise mximl subgrph of G tht hs no cutvertex. Lemm 2.2. G is plnr if nd only if ll its blocks re plnr. Proof. Let C be the set of cutvertices of G, nd B be the set of blocks of G. Let H be the block grph of G, whose vertex set is the disjoint union of B nd C, nd such tht block b nd cutvertex c re djcent if nd only if c b. This is biprtite grph which is esily seen to be forest; it gives corse description of G. For ech tree of this forest (corresponding to connected component of G), one cn trverse the tree, embedding ech block in turn without interfering with the other blocks. Lemm 2.3. Given grph G, we cn determine ll its blocks in liner time. Proof. We cn obviously ssume tht G is connected, becuse we could pply the lgorithm to ech connected component of G in turn. We rst focus on computing the cutvertices. For this purpose, run depth-rst serch on the grph G, strting from n rbitrry root vertex. Recll tht this prtitions the edges of G into link edges, which belong to the rooted serch tree T, nd bck edges, which connect vertex v with n scendent of v in T. Clerly, the root of T is cutvertex if nd only if it hs degree t lest two in T ; this property is trivil to test. It should be lso cler tht non-root vertex v is cutvertex if nd only if some subtree of T rooted t some child of v is incident to no (bck) edge whose other endpoint is n scendent of v. To test the ltter property eciently, during the depth-rst serch, we mintin the following informtion: the depth of ech vertex in the depth-rst-serch tree (once it gets visited), nd for ech vertex v, the lowpoint of v, nmely, the smllest depth of n endpoint of bck edge incident to descendent of v (possibly v itself), or if no such bck edge exists. The bove chrcteriztion indictes tht ( non-root vertex) v is cutvertex if nd only if the lowpoint of some child w of v is t lest the depth of v. Thus, we cn compute the cutvertices in liner time, provided we cn compute the depths nd the lowpoints in liner time. The depth is stndrd to mintin during depth-rst serch. The lowpoint of v cn be computed fter visiting ll descendents of v (i.e., just before v gets popped o the depth-rst-serch stck), since if we know the lowpoint of the children of v, we cn compute it for v in time liner in its degree. 9

10 ALGORITHMS FOR EMBEDDED GRAPHS 2. Plnrity testing nd grph drwing C mine the pieces of C. If C hs no piece, then G = C, thus G is plnr. If C hs two or more pieces, then C stises the conclusion, so we re done. So ssume tht C hs single piece P. Let v 1,..., v k be the ttchments of P on C, in cyclic order round C. Let p be pth in P between v 1 nd v 2. Now, let C be the circuit obtined by conctenting p with the subpth of C with endpoints v 1 nd v 2 tht lso contins v 3,..., v k (pick either of the two subpths if k = 2). One piece of C is the other subpth of C, nd nother piece of C is P \ p, unless P = p, in which cse G = C {p} is plnr. All of this tkes liner time. Figure 2.1. A grph G with circuit C (on the outside of the gure) nd the four pieces with respect to C numbered from 1 to 4. All pirs of pieces conict except (1, 3) nd (3, 4). There remins to explin how to compute the blocks. Notice tht, when processing v fter visiting ll descendents of v, every child w of v with lowpoint t lest the depth of v belongs to newly discovered block. For ech such w, we declre tht v, together with the connected component of G v contined in w, forms block, nd then we erse tht component of G v from the grph (to void tht block to be considered to be prt of new block lter). Lemms 2.2 nd 2.3 imply tht, for the proof of Theorem 2.1, we cn without loss of generlity ssume tht the input grph G is biconnected. Let C be circuit of G. We prtition the edges of G C into pieces s follows (see Figure 2.1): Two edges re in the sme clss if there is pth in G between them tht does not contin ny vertex of C. The vertices of piece P tht re in C re clled its ttchments. Since G is biconnected, ech piece hs t lest two ttchments. Lemm 2.4. In liner time, we cn either compute circuit of G tht hs t lest two pieces, or certify tht G is plnr. Proof. First compute ny circuit C, using, e.g., depth-rst serch. Deter- If G is plnr then, in plnr drwing of G, ech piece of circuit C must be entirely inside or outside C. We sy tht two pieces P nd Q of G re non-conicting with respect to C if, intuitively, in ny plnr drwing of G (if it exists), exctly one of P nd Q must be drwn inside C. More formlly, P nd Q re non-conicting if there re two (possibly identicl) vertices u nd v of C, splitting C into two subpths C 1 nd C 2 with endpoints u nd v, such tht ll ttchments of P re in C 1 nd ll ttchments of Q re in C 2. Otherwise, P nd Q re in conict. The conict grph of G with respect to C is grph with vertex set the pieces of C; two pieces re connected if nd only if they conict. Lemm 2.5. Let C be circuit of G. The grph G is plnr if nd only if the following conditions re stised: i. The conict grph of G with respect to C is biprtite; ii. for every piece P of G with respect to C, the grph obtined by dding P to C is plnr. Proof. Assume rst tht G is plnr. In plnr embedding, ech piece is drwn either entirely inside or outside C. Furthermore, two pieces P nd Q drwn on the sme side of C must be non-conicting becuse, in the cyclic order round C, edges of P nd of Q cnnot be interlced. (Otherwise, we would essentilly hve, fter removl, contrctions, nd expnsions of edges if needed, four vertices v 1, v 2, v 3, v 4 in this order on circuit C, with v 1 connected to v 3 nd v 2 connected to v 4 by edges inside C; dding new vertex outside C nd connecting it to ll four vertices, we would get K 5, 10

11 ALGORITHMS FOR EMBEDDED GRAPHS 2. Plnrity testing nd grph drwing v 1 () v 2 v 3 C Figure 2.2. () The circuit C (outside), nd some pieces embedded inside C. (b) After pushing the pieces of type i inside the regions bounded by C i nd U i, one cn embed the new piece P inside the disk formed by the pths U i. (These pths U i re not shown in the gure). which is nonplnr.) This implies tht the conict grph is biprtite. The second property is trivil. For the opposite direction, by (i), we consider biprtition P Q of the conict grph. We next describe how to embed ll pieces of P inside C; this concludes, since using similr method we cn embed ll pieces of Q outside C. We embed ech piece of P itertively. Assume tht we hve lredy embedded some pieces of P. Let new piece P hve ttchments v 1,..., v k in clockwise order long C; see Figure 2.2. For ech i, i = 1,..., k, let C i be the subpth of C tht goes clockwise nd connects v i to v i+1 (indices re tken modulo k). Ech piece of P lredy embedded hs its ttchments on subpth of C between v i nd v i+1 (in clockwise order), becuse it does not conict with P. In this cse we sy tht the piece hs type i. Let U i be pth just inside C with the sme endpoints s C i. Obviously we cn choose the U i s to be disjoint. Using suitble homeomorphism of the disk bounded by C, we cn push ll the pieces of P with type i into the disk bounded by C i nd U i while still hving n embedding. By (ii), piece P cn be embedded inside C; since its ttchments re v 1,..., v k, we cn push it inside the disk bounded by U 1,..., U k. In this wy, ll pieces re embedded nd disjoint, except possibly t the v 1 P (b) v 2 v 3 C ttchments v i. This shows tht piece P cn be embedded inside C, s desired. (Note tht pushing the existing pieces is not strictly needed, but it mkes the proof bit simpler.) At high level, the lgorithm rst pplies Lemm 2.4 to compute circuit C with t lest two pieces (unless G is plnr, which concludes). Then it uses the chrcteriztion of Lemm 2.5: If the conict grph of G with respect to C is non-biprtite, it returns tht G is non-plnr; otherwise, it recursively checks tht C P is plnr, for ech piece P of G (such grphs re clerly biconnected). The correctness is cler. To get n ecient lgorithm, however, we need to be slightly more specic. The lgorithm tkes s input biconnected grph G, nd circuit C of G with t lest two pieces. 1. Compute the pieces of G with respect to C. 2. Compute the conict grph of the pieces. If the conict grph is not biprtite, return non-plnr. 3. For ech piece P of G tht is not pth: () let G be the grph obtined by dding P to C; (b) let C be the circuit of G obtined from C by replcing the portion of C between two consecutive ttchments with pth of P between them; (c) pply the lgorithm recursively to grph G nd circuit C. If G is non-plnr, return non-plnr. 4. Return plnr. The correctness follows from the proof of Lemm 2.4 nd from the fct tht ech grph considered is biconnected. Now, we study the complexity of Step 2: Lemm 2.6. Given circuit C, we cn determine the conict grph of G with respect to C in qudrtic time. Proof. Let P be piece of C, with ttchments v 1,..., v k in cyclic order round C. Then nother piece Q does not conict with P if nd only if ll its ttchments re in some intervl [v i, v i+1 ], in cyclic order round C 11

12 ALGORITHMS FOR EMBEDDED GRAPHS 2. Plnrity testing nd grph drwing (indices re tken modulo k). This suggests the following pproch: Mrk ech vertex of C ccording to which intervl(s) [v i, v i+1 ] it belongs to; for ech piece Q P, determine if ll its ttchments belong to single intervl using this mrking. This tkes liner time plus time liner in the number of ttchment points of ll the pieces, which is lso liner. Iterting for every piece P, we obtin the conict grph of G in qudrtic time. Since testing whether grph is biprtite cn be done in liner time, this shows tht ech recursive invoction of the lgorithm tkes qudrtic time. Furthermore: Lemm 2.7. The number of recursive invoctions is liner in the complexity of the input grph. Proof. We ssocite dierent edge of G to ech invoction of the recursive lgorithm. Nmely, for given invoction on grph G nd circuit C, we select witness edge e of C tht does not belong to the circuit of the prent invoction. Tht edge e does not pper in the siblings' grphs, so it will not show up s witness edge in ny sibling invoction nor in ny descendent of sibling. There remins to prove tht e does not pper s the witness edge of descendent invoction. Wlk in the recursion tree towrds tht descendent. While e belongs to the circuit of the invoction, it cnnot be chosen s the witness, since it belongs to the circuit of its fther. When e ceses to belong to the circuit of the invoction, then by choice of the new circuit C, e now belongs to piece of C tht is pth, nd therefore is bsent from ny descendent invoction. This proves Theorem 2.1 with weker, cubic-time complexity... well, ctully not quite: We only determined whether the input grph is plnr or not; in the former cse, little bit more work is needed to ctully compute combintoril mp: Exercise Convince yourself tht one cn, lso in cubic time, compute n embedding if the input grph G is indeed plnr. 2.2 Grph drwing on grid Now we consider the following problem: Given plnr grph G, given in the form of combintoril mp (for exmple, obtined by the lgorithm in the previous section), how cn we build n ctul embedding of G in the plne? To be more specic, we need some denitions. A simple grph is grph without loops or multiple edges. A plnr grph is tringulted if every fce of G, including the outer fce, hs degree three. A grph embedding in the plne is stright-line if every edge is stright-line segment (such n embedding is thus uniquely determined by the coordintes of its vertices). We shll prove: Theorem 2.9. Let G be simple plnr grph, given in the form of combintoril mp. In O(n) time, we cn compute stright-line embedding of G where the vertices re on regulr O(n) O(n)-grid. The restriction of hving simple grph is legitimte, becuse non-simple grphs do not hve stright-line embedding. Furthermore, we cn remove ll loops nd multiple edges in grph in liner time if desired: Lemm Let G be grph (not necessrily plnr) of complexity n. In O(n) time, we cn determine ll loop edges nd multiple edges of G. Proof. Let v be vertex of G. Mrk ech neighbor w of v with the list of edges with endpoints v nd w, by visiting ech edge incident with v in turn. Any list contining more thn one edge corresponds to multiple edges; if the list of v is non-empty, it corresponds to one or severl loops. Finlly, we erse the mrks on the neighbors of v. This opertion tkes time liner in the degree of v. We cn iterte the process over ll vertices v in turn. Reusing the technique, we lso obtin: Lemm Let G be simple plnr grph, given by its combintoril mp. In liner time, we cn dd edges to G to obtin simple, tringulted, plnr grph, lso given by its combintoril mp. 12

13 ALGORITHMS FOR EMBEDDED GRAPHS 2. Plnrity testing nd grph drwing Proof. We rst mke G connected. Let G 1,..., G k be the connected components of G; for ech i, 1 i k 1, we dd n edge e i connecting n rbitrry vertex v i of G i with n rbitrry vertex v i+1 of G i+1, inserted rbitrrily in the cyclic orders of the edges round v i nd v i+1. The resulting grph G cn be embedded in the plne (or sphere), nd cellulrly becuse G is connected. In such n embedding, ech fce of G is homeomorphic to n open disk, nd hs degree t lest three becuse there is no loop or multiple edge. We now dd edges to G in order to mke it tringulted. For this purpose, for ech fce f of degree d 4, we choose n rbitrry vertex v incident to f nd tringulte f by dding d 2 edges in f with v s one endpoint, nd thus replcing f with d 2 tringles. This involves modifying O(d) gs for fce f, so tkes liner time in totl. The only problem is tht this opertion might hve creted loops nd multiple edges (for exmple, if v ppered t lest twice on the boundry of f, this opertion cretes loop bsed t v). Our lgorithm will remove loops nd multiple edges by itertively ipping some loop nd multiple edges tht were not in the originl grph G; ipping edge e mens removing it, trnsforming the two incident tringles with single qudrngle of which e ws digonl, nd dding the edge tht is the other digonl of the qudrngle. Flipping n edge tkes constnt time. Let us rst eliminte the loops bsed t vertex v. A hlf-edge is n incidence between such loop edge nd v; thus, every loop edge bsed t v corresponds to two hlf-edges. Consider the cyclic order of the hlfedges round v (we emphsize tht, in this cyclic order, we ignore the nonloop edges). There must be n edge whose two hlf-edges re consecutive round v. (Proof: consider n edge e whose hlf-edges re closest in this cyclic order. These two hlf-edges seprtes the cyclic sequence of hlfedges into two liner sequences. If the shortest of these sequences is not empty, it contins the hlf-edge of some edge e. The other hlf-edge of e must lso be in this sequence, by the Jordn curve theorem. This contrdicts the choice of e.) The edge e obtined by ipping e is not loop; indeed (Figure 2.3, left), the reltive interior of e crosses e exctly once, so by the Jordn curve theorem, the endpoints of e must be distinct, except if they re both equl to v; but t lest one of the endpoints of e is v v Figure 2.3. Left: Flipping loop with no loop inside it gives non-loop. Right: If tringulted plnr grph hs no loop, then ipping multiple edge does not crete loop or multiple edge. dierent from v, by our choice of e. Thus, our lgorithm itertively nds loop edge bsed t v whose two hlf-edges re consecutive in the cyclic order of the loop edges round v nd ips it. (No such edge belongs to the originl grph G.) To do this in totl time proportionl to the degree of v, we initilly compute the list of ll loop edges bsed t v whose hlf-edges re consecutive, nd updte this list s we ip edges. Building this list tkes time liner in the degree of v, nd updting it tkes constnt time per updte (by mintining the cyclic order), whence the complexity. Iterting the bove procedure to ech vertex v, we cn ssume tht the grph hs no loop. Let us now explin how to eliminte the multiple edges incident to given vertex v. For ech neighbor u of v, consider the set of edges E uv with both u nd v s endpoints. We cn compute E uv using the technique of the previous lemm. Assume E uv 2. The originl grph G hs t most one edge in E uv ; if G contins one edge of E uv, we let e be tht edge, otherwise we let e be n rbitrry edge of E uv. Now ip ll edges in E uv \ {e}. No ipped edge cn be loop or multiple edge, by plnrity of the tringulted grph nd becuse there is no loop before the ip (Figure 2.3, right). Iterting this process for ech vertex v in turn, we obtin the desired liner-time lgorithm. The previous lemm implies tht, to prove Theorem 2.9, we cn ssume tht G is tringulted. Another key ingredient for the proof of this theorem is the following inductive decomposition of plnr, simple, tringulted grph, depicted in Figure

14 ALGORITHMS FOR EMBEDDED GRAPHS 2. Plnrity testing nd grph drwing 7 w i+1 w j w i v 1 v 2 3 Figure 2.5. Illustrtion of the proof of Lemm Figure 2.4. Illustrtion of Proposition The directed tree is used lter in the proof of Theorem 2.9. Proposition Let G be plnr, simple, tringulted grph. Let v 1 nd v 2 be two vertices on its outer circuit. In liner time, we cn order the vertices of G s v 1,..., v n such tht, for ech k 3, the subgrph G k of G induced by v 1,..., v k stises: G k is connected; the boundry of G k is circuit; ech inner fce of G k hs degree three; v k+1 is in the outer fce of G k. The proof of this proposition rests on the following lemm. Lemm Let G be plnr, simple grph; ssume tht the boundry of the outer fce forms circuit (without repeted vertices) C. Let v 1 v 2 be n edge on C. There exists vertex v on C, dierent from v 1 nd v 2, tht hs exctly two neighbors on C. Proof. If every vertex of C hs exctly two neighbors on C, we re done. Let the vertices of C be v 1 = w 1,..., w m = v 2, in this order. Consider n edge connecting w i to w j where j i is miniml but t lest two. Then the only neighbors of w i+1 in C re w i nd w i+2 (Figure 2.5): None of w i+3,..., w j cn be neighbor of w i+1 by minimlity of j i, nd none of the other vertices on C either, by plnrity. Proof of Proposition We choose v n,..., v 3 in this order by repeted pplictions of Lemm 2.13; the conditions re obviously stised. To do this in liner totl time, we mintin the following informtion on ech vertex v of the current grph: Whether v belongs to the outer circuit nd, if so, its number of neighbors on the outer circuit. We mintin list of (pointers to) vertices on the outer circuit tht hve exctly two neighbors on the outer circuit; by Lemm 2.13, this list is never empty. The lgorithm itertively picks vertex in the list, updtes the dt, nd itertes until exctly three vertices re left. This tkes liner time, since ech edge is considered only if one of the endpoints enters or leves the circuit. Proof of Theorem 2.9. The lgorithm itertively embeds the subgrph G k of G induced by v 1,..., v k, where k goes from 3 to n. Actully, insted of computing x- nd y-coordintes of the vertices, we compute y-coordintes of the vertices nd x-spns of the edges, nmely, the dierence between the x-coordintes of their endpoints; trivilly, this informtion is enough to recover the embedding. 14

15 ALGORITHMS FOR EMBEDDED GRAPHS 2. Plnrity testing nd grph drwing v 1 = w 1 w p w p w p+1 w p+1 P (w p, w q ) w q 1 P (w p, w q ) w q 1 v 1 = w 1 w m = v 2 w q w q w m = v 2 Figure 2.6. Illustrtion of the proof of Lemm Assume inductively tht we lredy embedded G k (k 3) on the grid in such wy tht (Figure 2.6): 1. The y-coordintes of v 1 nd v 2 re zero; 2. If v 1 = w 1,..., w 2,..., w m = v 2 re the vertices on the outer fce of G k, in cyclic order, then the x-spns of ech edge w i w i+1 is positive; 3. ech edge w i w i+1, 1 i m, hs slope +1 or 1. Vertex v k+1 is incident, in G k+1, to contiguous set of vertices w p,..., w q on the boundry of the outer fce of G k. Let P (w p, w q ) be the intersection point of the line of slope +1 pssing through w p with the line of slope 1 pssing through w q ; Condition (3) implies tht P (w p, w q ) hs integer coordintes. Putting v k+1 t position P (w p, w q ) lmost yields plnr drwing of G k+1, except tht it my fil to see, e.g., w p. To void this problem (Figure 2.6), we shift vertices w 1,..., w p by one unit to the left, so tht the slope of w p w p+1 becomes now smller thn +1; nd similrly we shift w q,..., w m by one unit to the right. In our choice of representtion of points with x-spns nd y-coordintes, this tkes constnt time: Simply increse by one the x-spn of w p w p+1 nd of w q 1 w q. The only problem is tht the resulting drwing is inconsistent, so we need n djustment phse to increse the x-spns of some internl edges. We rst explin how to do this djustment of the x-spns of internl edges t ech step from G k to G k+1. However, for the purposes of n ecient lgorithm, it will be useful to do these djustments t once. We mintin spnning tree T of the dul of G k, rooted t the outer fce nd oriented wy from the root, s follows (Figure 2.4). Initilly (sy k = 3), there is one edge from the root outer fce to the inner fce, crossing edge v 1 v 2. When we dd vertex v k+1, for ech newly creted internl fce of the drwing, we crete n edge of T rriving to tht fce by crossing the unique edge incident to tht fce tht belongs to G k. When dding edges in G k to build G k+1, the djustment phse consists in incresing by one the x-spn of the set E p of edges crossed by the subpth of T from the root to the rst vertex incident to (w p w p+1 ), nd similrly of the edges E q 1 crossed by the subpth to the rst vertex incident to (w q 1 w q ). (Edges crossed by both subpths hve thus their x-spn incresed by two.) Combined with the initil shift of the boundry edges, this results in shift of left prt of the grph to the left nd of right prt of the grph to the right. Why does this result in vlid embedding? It suces to prove the following stronger result by induction on k: If the outer fce of G k is w 1 w 2... w m, then for ny choice of positive integers δ 1,..., δ m 1, if for ech i we increse the x-spn of the edges in E i {w i w i+1 } by δ i, then we obtin n embedding. This is esy to prove for k = 3; proving it for k mounts to proving it for k 1 (for well-chosen dierent vlues of the integers) nd to checking tht the new edges in G k do not cross ny other prt of the drwing (by construction). It is cler tht, t the end, the vertices re on n O(n) O(n)-grid. To implement this ide in liner time, we rst compute the x-spns nd y- coordintes in G 3,..., G n without doing the djustment phses; this tkes O(n) time. Omitting this djustment phse does not hrm becuse, t ech step, we only need to know tht the x-spns nd y-coordintes of the vertices on the outer fce re correct. Afterwrds, we need to increse the 15

16 ALGORITHMS FOR EMBEDDED GRAPHS 2. Plnrity testing nd grph drwing x-spn of ech edge e by the cumulted increse it would hve received during ll djustment phse. This mounts to determining how mny times e is crossed by the pths of T considered during the djustment phse. For this purpose, during the incrementl construction, we record, for ech vertex of T other from the root, the number of times it ppers s n endpoint of such pth. At the end of the incrementl construction, we cn by simple serch in T compute, for ech edge of T, the number of times it is contined in pth. This tkes liner time. 2.3 Tutte's brycentric embedding theorem We give nother method to build stright-line embeddings in the plne, which hs other desirble properties: In prticulr, ssuming 3-connectivity of the input grph (dened below), in the plnr output embedding, ll fces re convex. This leds to n interesting result regrding 3-dimensionl polytopes, described in the next section. A grph is 3-connected if it is connected nd if it is still connected fter removing zero, one, or two vertices nd their incident edges. Theorem Let G = (V, E) be 3-connected grph without loops or multiple edges. Assume G is embedded on R 2. Let v 1,..., v k be the vertices of the outer fce. Assign unique positions f(v) in R 2 for ech vertex v, such tht the f(v i ), i = 1,..., k re mpped to the vertices of convex polygon (respecting the order of the vertices); the imge of every vertex v dierent from the v i 's is brycenter with strictly positive coecients of the imges of its neighbors in G. Then drwing stright-line edges between the imge points gives n embedding of G. Given such 3-connected grph, it is lwys possible to chieve the conditions of the theorem. For exmple, choose the brycentric coecients to be ll equl to one. The brycentric condition yields n ne system, which is solvble by n rgument of dominnt digonl. Equivlently, Figure 2.7. Illustrtion of the sttement of Tutte's theorem. one my view the edges s springs with the sme rigidity, nd the interior vertices s being free to move. The equilibrium of this physicl system is met when the energy is minimized [74, p. 124]. We refer to the vertices v 1,..., v k s exterior vertices, nd to the other ones s interior vertices. Let v be n interior vertex of G. Let h : R 2 R be n ne function vnishing on f(v). If ll the neighbors of v lie on h 1 (0), we sy v is h-inctive. Otherwise, v is h-ctive. In this cse, v hs neighbors in both h 1 ((0, )) nd h 1 ((, 0)). In prticulr one cn nd rising pth from v to n exterior vertex: pth whose vlue of h strictly increses. Similrly one cn nd flling pth. Proposition The imge of every interior vertex of G is in the interior of the convex polygon f(v 1 )... f(v k ). Proof. Let h be n ne form such tht the polygon f(v 1 )... f(v k ) lies in h 1 ((0, )). If there is vertex (whose imge is) in h 1 ((, 0)), then consider the one tht hs minimum vlue of h. Since it is brycenter with positive coecients of its neighbors, ll its neighbors must hve the sme vlue of h. By induction nd connectivity of G, some exterior vertex must hve tht vlue of h, which is not possible. Therefore, ech interior vertex lies in the interior or on the boundry of the polygon f(v 1 )... f(v k ). Let v be n interior vertex; ssume v lies on n edge of the outer polygon, whose supporting line is h 1 (0). Then ll the neighbors of v re h-inctive. Thus, ll interior vertices tht cn be reched from v by pth using only interior vertices lie on h 1 (0). This contrdicts the 3-connectivity of G, becuse removing the two exterior vertices on h 1 (0) destroys the 16

17 ALGORITHMS FOR EMBEDDED GRAPHS 2. Plnrity testing nd grph drwing w 1 w 1 w 2 v w 2 v v w 3 connectivity of G. Figure 2.8. The sitution in the Ylemm. w 3 v Proposition For ny ne form h, there is no h-inctive vertex. See Figure 2.8 for n illustrtion of the next lemm. Lemm 2.17 (Ylemm). Let w 1, w 2, w 3 nd v be pirwise distinct vertices of grph H. Assume, for i = 1, 2, 3, tht there is pth P i from w i to v which voids the w j 's (for j i). Then there exist three pths P i, from w i to common vertex v, which re pirwise disjoint (except t v ). Proof. First, using P 1 nd P 2, we esily get (simple) pth R from w 1 to w 2, so tht R nd P 1 hve the sme rst edge w 1 z. Then we consider the pth P 3. If this pth P 3 intersects R, let v be the rst vertex of intersection on P 3. v splits R in two prts, which we cll P 1 (from w 1 to v ) nd P 2 (from w 2 to v ); P 3 is the prt of P 3 going from w 3 to v, with loops removed (if ny). The P i 's stisfy the property stted in the lemm. If P 3 does not intersect R, we cll v the lst vertex on P 1 (when going from w 1 to v) which is lso on R. Such vertex exists nd is dierent from w 1 becuse w 1 z is the rst edge of R nd P 1. Let P 3 be the pth dened by P 3 followed by the prt of the pth P 1 which goes from v to v, with loops removed (if ny). v splits R in two prts, which we cll P 1 nd P 2. The pths P i 's stisfy the desired property. Figure 2.9. A summry of the proof of Proposition Proof of Proposition For the ske of contrdiction, ssume v is n h-inctive vertex. We prove the existence of subdivision of K 3,3 in G: subgrph of G such tht, fter contrcting edges, we get K 3,3. Thus G cnnot be plnr. See Figure 2.9. Let G(h) be the subgrph of G induced by the vertices on h 1 (0). Since G is 3-connected, there re, in G(h), three distinct h-ctive vertices w 1, w 2, nd w 3 nd three pths P i connecting v with w i, such tht, for ny i, the pth P i contins no vertex w j for j i. Indeed, let w be vertex of G so tht h(f(w)) 0. By connectivity of G, tke pth from v to w nd cll w 1 the rst h-ctive vertex on this pth. Do the sme in G {w 1 }, nd choose w 2, by 2-connectivity. Similrly, use 3-connectivity to select w 3 in G {w 1, w 2 }. Applying then the Ylemm in G(h), we get the existence of vertex v in G(h), together with three distinct pths (except t v ) P i from w i to v in G(h). We cn build rising pths Q i from ech of the w i to vertex x mximizing the vlue of h. Then, the Ylemm llows us to ssume, by chnging x nd the Q i 's if necessry, tht these three pths re disjoint (except t x). Similrly, we cn build flling pths R i from ech of the w i to vertex y nd pply the Ylemm. Using the pths P i, Q i nd R i, which re ll pirwise disjoint except t their endpoints, nd the vertices x, v, y nd w 1, w 2, w 3, we get subdivision of the grph K 3,3. This contrdicts the plnrity of G. By Proposition 2.16, the convex hull of the neighbors of n interior vertex v is non-degenerte polygon, nd v lies in its interior. We now tringulte 17

18 ALGORITHMS FOR EMBEDDED GRAPHS 2. Plnrity testing nd grph drwing Figure Two cses for the proof of Lemm Vertices u nd v, together with flling pths, re highlighted. the fces of G, except the outer fce: fces re dded to split the fces of G into tringles, without dding vertices (this is done in purely combintoril wy). It still holds tht v is in the interior of the convex hull of its neighbors, which redily implies tht v is brycenter with positive coecients of them. To summrize, the hypotheses of the theorem re stised, but we cn now even ssume tht every fce of G, except mybe the outer fce, is tringle. Proposition Let uvy nd uvz be two tringles of G shring the edge uv. Let h be n ne form vnishing on f(u) nd f(v). Then h(f(y))h(f(z)) < 0. The proof relies on the following lemm. Lemm If h(f(y)) > 0, then h(f(z)) < 0. Proof. By ssumption u, v, nd y re h-ctive. Find strictly flling pths going from u nd v to n exterior vertex. The flling pths my shre vertex. In ny cse, we get simple circuit C in G using uv whose set of vertices re on the strictly negtive side of h, except u nd v. See Figure (We my need some exterior vertices if the flling pths do not shre vertex.) On the originl (not necessrily stright-line) embedding of G, the circuit C bounds disk. Let S be the set of vertices in the interior of this disk; S contins no exterior vertices, so every vertex in S is brycenter with positive coecients of its neighbors. Under f, ll the vertices of C re mpped to the hlf-spce h 0 except u nd v, which re mpped to h = 0; therefore, s in the proof of Proposition 2.15, ll the vertices in S belong to the open hlf-spce h < 0. Since h(f(y)) > 0, the vertex y cnnot belong to C or S. In the originl embedding of G, the circuit C uses edge uv, nd y is outside C. Therefore z must be inside C (i.e., in S) or on C. In the former cse, s seen bove, we get h(f(z)) < 0. In the ltter, since u nd v re the only vertices of C on the line h = 0, we lso get h(f(z)) < 0. Proof of Proposition Lemm 2.19 shows tht, whenever one tringle is non-degenerte, then its incident tringles re non-degenerte. Necessrily, ny tringle hving one exterior edge is non-degenerte (Proposition 2.15). So every tringle is non-degenerte. The result follows. We cn now conclude the proof of Tutte's theorem. Proof of Theorem As we discussed erlier, consequence of Proposition 2.16 is tht we my ssume tht G is tringulted (except possibly for the outer fce). Since the tringles re non-degenerte by Proposition 2.18, it suces to prove tht the interiors of two distinct tringles re disjoint. For the ske of contrdiction, let be point of R 2 in the interior of two tringles t nd t. Shoot ry from to the boundry of the polygon f(v 1 )... f(v k ) voiding the imge of every vertex. Whenever the ry leves t, by Proposition 2.18, it enters nother tringle. So we get sequence of tringles t = t 1, t 2,..., t where t is the unique tringle incident to the boundry edge tht is on the end of the ry. Similrly, we get sequence of tringles t = t 1, t 2,..., t. Going bck in both sequences from t, we pss from tringle to n unmbiguously dened preceding tringle. Since we strt with the sme tringle, we get t = t. In prticulr, we hve: Corollry 2.20 (FárySteinWgner's theorem). Every plnr grph cn be drwn in the plne with stright line edges. 18